I want my world to be set in early to mid Industrial Age but for various reasons that I won't go into here I don't want there to be any gunpowder based weapons. Is it possible for a society to advance to the Industrial Age without ever discovering gunpowder?
It is certainly possible. Mechanisation is what defines the industrial age, and mechanisation is entirely separate to gunpowder. The first thing that was mechanised was agriculture with things like reaper-binders. This freed enough population from working on the land to be employed in factories. The next was the textile industry which was relatively simple to automate and from there it snowballed.
There is however a major caveat to all of this. As your industrial revolution progresses more and more chemistry is going to be required for things like metallurgy (which is fairly important if you want metal good enough for things like steam engines.) The further into an industrial revolution the less and less likely that a simple chemical mixture like gunpowder would remain undiscovered.
It's quite hard to say. Drilling and blasting has been key to mining and quarrying since 1627 and because mining is key to industrialisation it's certainly going to have an effect.
Whether it would prevent a culture from industrialising or not depends on how well that culture manages to bypass explosive mining, which is still standard practice, and develop mechanical mining processes to replace it.
Maybe it is not nice to say, but warfare and weapon research is often the main driver for techical development.
Just to give an example in your reference era, thermodynamic started when somebody noticed that drilling a cannon generates heat and then started investigating this.
If you remove gunpowder you remove some if not all of the need for better metals, than can withstand the explosion without blowing off (too often) the face of the gunman.
Also, no gunpowder would have made the expansion of Western nations into America, Asia and Australia more difficult if not impossible, due to the limited logistic capabilities which could not sustain a long term engagement far away from homeland. This would in turn have reduced the growth of those nations and, again, the need for better technologies.
Therefore my answer is no, without gunpowder it wouldn't be possible for a society to develop until industrial revolution.
There are two problems one will have to solve, to allow industrial technology but no gunpowder: firstly, how could an industrial society not discover gunpowder on its own? and secondly, could an industrial revolution (or a slower transition towards industrialization) happen without gunpowder?
problem 1: Not allowing gunpowder development
(edit) I understand that the main point of the question is to examine the effects on the industrial society, and not explaining why gunpowder couldn't be developed, but I do still consider this relevant since the reason for gunpowder not to be developed still would affect the society.
The first thing one could do to make gunpowder development impossible would simply be to claim that the components of gunpowder (coal, saltpeter ($KNO_3$), and sulfur) are very rare on the planet on which this civilization exists; this explanation would, however, have some problems since the chemical elements involved in making gunpowder are essential building blocks of living organisms, because one would have to come up with an explanation for how this planet ended up without some of these elements, and because a scarcity of coal would make industrialization quite hard.
Not even to mention that the chemists of your society still could invent other explosives, which doesn't include the rare components.
If for instance sulfur is rare, you should still be able to make nitroglycerin and TNT, and if it is nitrogen which is rare, oil (both vegetable oil and from the underground) could be used to make firebombs.
Maybe the dominant culture or religion could ban explosive weapons, perhaps because they consider it to be demonic, or out of fear that the population would use this weapon against the ruling class.
There are an obvious problem with this: if one civilization bans gunpowder, it would easily be overtaken by another one, which doesn't. Using this explanation might, therefore, require the assumption that one civilization long before gunpowder was discovered, became the dominant power on the planet – If you want this to be set on earth, you could for instance claim that Alexander the Greats empire, the Persian empire, Rome, Carthage or one other ancient power came to dominate the world.
This dominant civilization would not be threatened by any other power in the world, in would therefore not need powerful weapons such as gunpowder for military purpose, and with the influence that comes with being a superpower, this dominant empire might be able to force any still independent nations to also obey their gunpowder-ban.
The superpower could then go on to eventually (possibly much later than if they had had gunpowder, and as pointed out in a comment, maybe not as a revolution but rather a slow transition) invent industrial technology on its own, and even if the superpower at some point should collapse, leading to the end of the gunpowder-ban, it would still take a while before anybody developed effective gunpowder weapons.
Problem 2: industrialization without gunpowder
Something many people have pointed out is that gunpowder has been used in mining for a very long time, and one could therefore validly have doubt to if a society without use explosives could mine enough resources to fuel an industrial society: But it is actually possible to do explosive mining without any explosive chemicals. The Roman empire did, for instance, use the Ruina Montium technique, in which they would lead a lot of water into a mountain very quickly, causing it to explode from the pressure.
I do of course not know if Ruina Montium alone could substitute explosives in mining, but at least the technique shows what an already advanced civilization can do without gunpowder.
In conclusion, I can neither say yes nor no, to the question: »Is it possible to have a Society to evolve into an industrial age society without discovering gunpowder?«. I have tried to outline some assumptions one could make, but I am not even convinced myself that they would be enough to achieve what you want, but even if it is possible I find it quite unlikely that an industrial society would not have gunpowder
I have only now noticed that another answer posted about the time when I started writing this answer contains the same suggestions for explanations as to why gunpowder couldn't be developed, this answer was however written completely independent from that other answer, and i believe it looks at thing from a slightly different angle, but if this answer is to similar and ought to be removed, please write that in the comments.
The answer is not really.
Here is the thing: Industrialization demands some kind of advanced chemistry. And in chemistry, you end up with explosives very, very quickly and straight forward. It's like asking if humans as a species could learn swimming without discovering fish - they are right there in front of you. There is just no way that we could've had anything close to our industrialization without discovering any explosives.
a) If there is little war, explosives might have not ended up on the battle field
b) If you would change certain aspects, making the industrialization "more primitive" - look, this needs to be a new question because this post would go on for pages, also please do some research yourself if you end up asking that question - if you drop all the things one can only do without chemical knowledge - you might reach your goal
If I understood the question correctly, No and Yes.
No, it is extremely unlikely for an industrial civilization to not know and use gunpowder.
It is a simple mixture of three fairly common and easily accessible substances. It is unlikely that discovery of gunpowder could be prevented by its components being scarce. It is also unlikely that a civilization capable of industrialization would fail to have the necessary chemistry and notice the effect saltpetre has on burning. Or fail to use explosives for mining.
Yes, it is possible for them to never use the gunpowder for guns. Making it, I guess, blasting powder instead. Or black powder. Or fire powder.
There is a direct link from discovering black powder to fireworks and low explosives, but guns actually require an extra step IMHO. Your people need to want siege weapons capable of breaking walls. If your civilization doesn't have fortifications it doesn't need siege weapons. If it doesn't need siege weapons, it will have no need for cannons. If it doesn't have cannons, it won't downscale them to guns. No airguns either.
Having fortifications and being industrialized do not seem to be connected in any way as far as I can tell. Thus there should be no reason not to have an industrialized civilization without fortifications.
Of course, the history of such civilization would be quite different from ours. How different is really a separate question, but I think your world would need to be dominated by strong and stable central governments capable of suppressing rebellions and unrest without needing fortifications to protect themselves. A government that has no use for fortifications would naturally ban anyone else from having any. And really, fortifications are too expensive to build and upkeep unless you actually need them.
Additionally, you'd need the government to not need to fear external threats. Some empires did actually reach this level of power for a period of time. There are several scenarios for this. You can have a single superior empire that suppresses "serious" warfare in its entire area of influence. You could have a diplomatic system between states that suppresses warfare. Either in general or merely siege warfare. Siege warfare makes wars longer and more expensive, so people could simply agree not to build fortifications if conditions are right. Such agreement would probably only last until the next major war, but it would suppress development of siege weapons. And the political system could be stable enough to prevent major wars.
Yet another alternative is to have a weapon more effective than primitive guns. Remember the first guns were muzzle loaded balls with a packet of gunpowder, the barrels were not rifled to create spin so the guns were wildly inaccurate. The parts were not interchangeable, either, a broken gun meant making a new part by hand. Plus they could literally blind and deafen the shooter!
So the 'alternative history' could be to advance industrialization well before the discovery of gunpowder.
IRL, the Viking's had access to crucible steel 1200 years ago; we see it in some of their swords (Ulfbhert, see http://www.pbs.org/wgbh/nova/ancient/secrets-viking-sword.html, also http://www.dailymail.co.uk/sciencetech/article-2878512/The-mystery-magical-superstrong-Viking-sword-Researchers-close-supermonks-believed-forged-weapons.html). This is steel as good and hard as modern steel, and available hundreds of years before the first written formula for gunpowder.
So, alternate history wise; suppose instead of just making swords with it, it was forged into machinery, and the magicians that made the crucible steel figured out how to make some of Archimedes weapons, too: repeating crossbows, lathes with diamond tips making fine machinery, spring steel that let archers fire accurately for hundreds of yards: Far further and more deadly than the best early blunderbusses. Suppose they fired needles, filled with lethal poison, and could actually be hand held like pistols, but with a hundred rounds instead of 6 of those big bulky bullets.
Our firearms have slowly evolved to be accurate and lethal at a distance, but that is for the same reason we have the QWERTY keyboard: It came first, was widely adopted, and became the awkward norm. If industrialization starts 300 years before we discovered gunpowder circa 1100 AD, because for some cultural quirk everybody knows how to make good steel, and some Archimedes or Da Vinci type figure of the time could make kinetic weapons powered by spring steel that made early gun shooters look like idiots playing with dynamite for no good reason; the bullets did not travel as far or as accurately as their steel darts: Say a marksman with a steel dart could hit the bullseye three times in 8 seconds at 50 yards, while a blunderbuss takes two minutes to load one shot and would be lucky to hit the target at all!
Modern steel, the discovery of fine cutting and grinding with diamonds, the invention of standardized parts. Metallurgy of our many kinds of steel would be obvious; the super hard alloys for scalpel sharp edges can happen.
On top of all that, industrial production of modern steel would help with the exploitation of other kinds of explosive fuels: Natural gas, gasoline and internal combustion engines, nitroglycerin. Even electrical discoveries are plausibly fuelled by fine metallurgical advances and refinement of metals and experiments with alloys.
Instead of gunpowder, magnetic rail guns.
The writing trick is just that something else got there first by a few centuries. It was evolved and refined by hundreds of little inventions and improvements, it was widely accepted in the culture, and the headstart was just too much for gunpowder to overcome: People were accustomed to "charging" their guns with machinery and/or horses winding thousands of pounds of energy into reliable heavy steel springs: Their guns were whisper silent, deadly, accurate, safe, carried dozens of projectiles and left no plume of smoke identifying the location of the shooter! Which made the first black-powder guns look like, to them, cavemen throwing rocks! It isn't that black powder and other such explosives were never discovered, just that the black powder gun was stillborn, a loud, smelly, inaccurate novelty, handmade by a few crackpots, that nobody ever saw fit to improve upon.
One could make the argument that warfare drives technological advancement. The military budget is among the largest of the federal budgets, and they often fund research and get first dibs on potential applications. As an aside, consider that the US hasn't put astronauts on the moon recently because there is no cold war driving the effort. Any non-military applications are unintended (like mri machines). Even Einstein, who signed a letter to the US President at the time to not drop nukes, had his discovery of mass-energy-equivalence primarily used for military purposes, the studies of which further contributed to science driving military applications.
That said, it is not necessarily the case that only military funding drives technological advancement. But I struggle to find a continual funding source to invest in non-military applications that help further the cause. As a hypothetical, consider that the most dominant countries will have the most funding available for science research and also prioritize military spending to maintain said dominance.
A large military budget. When guns were first developed they were hugely worse than a bow and arrow. However they were much easier to train a soldier to use. If bows had been more popular then the production of guns would not have improved.
The cannon was used because before the cannon an enemy force could hide in a castle until they were ready to fight. A standing army would remove the need for castles, and thus early cannons.
I would say Yes. If you are simply not looking for gunpowder, but allow other explosives, Dynamite which was largely used for mining, gets its explosive power from nitroglycerin, which has nothing to do with gunpowder, but would allow your civilization to mine effectively. If another concern is warfare, an often overlooked weapon is the airgun, which has the advantages of being silent, fireable during wet weather, does not produce smoke and can be fired rapidly depending on how much compressed air you have. They just tend to be weaker comparatively, but could defiantly be a driving force in warfare to spur development. You could also use pneumatically launched dynamite as a type of artillery.