Problem: 1/12th size is not 1/12th power or force
Their main problem, as noted in the other answers (or comments to them), is that force/power does not scale linearly with size. A 6 inch humanoid may be able to carry a toothpick-sized spear, but it won't do much with it - it just won't have the strength to pierce even skin, let alone fur+skin+fat on a rat or similarly-sized animal trying to eat it.
Using brains and teamwork to replace muscles
Now we're thinking smart little humanoids. There ARE ways to get around that issue with enough effort and knowledge. A group of Lilliputians could concievably operate a complex system of levers, pulleys and gears to wind up a scorpio (spear-throwing ballista-like siege engine) that would look like a small crossbow to a normal-sized human. With this, they could fend off smaller predators - hopefully they're beneath the notice of any bears. Mechanical systems like that would also allow them to handle fuel and ores in quantities that make using metals feasible.
So the question is then, do you allow them to start with technology (or at least knowledge of it) far beyond your average stone-age person? Or if not, can they somehow survive long enough and produce excess food enough to afford thinkers to tinker until they can come up with the ideas and principles, and do all the work needed to create the first machine that goes beyond a curiosity?
Pulley systems would be somewhat easy to discover - at their size, a single plant fiber or some discarded animal hair would make a good base for a "rope", and three conveniently grown tree branches (polished with pebbles to reduce friction) could get them a 2:1 pulley system. A group of Lilliputians tying a rock to a rope, pulling it up as a team effort, and dropping it on a walnut might be able to crack it more easily than one little guy trying to carve into it with a teensy tiny hand axe for days. If they manage to scavenge up some bones, they could get to their marrow more easily this way too. So there's incentive and reward for discovering basic mechanics, which might make them view tinkers favorably and invest into new ideas too.
Levers too are fairly obvious - a twig with a rock as fulcrum may let your group of gatherers turn over a rock and get at the tasty worms and grubs just below it.
Gears are more challenging - you'll need a stationary settlement to really make use of them. Other commenters note that without sufficient protection against rats, foxes, cats or what have you, such a settlement would be seen as larder for any predator who notices them. If the Lilliputians are nomadic prior to the discovery of enough mechanical knowledge and skills to fend off small predators, this might be an impediment - but not an insurmountable one.
A bit of culture to make it work
What won't work is the fairy tale idea of one guy sitting down, whittling up some gears and rods and other pieces, and putting it all together in a tribe-protecting mini siege engine. Even if we let Lilliputians live human-long lives rather than the much shorter ones they SHOULD have at that size, and then replace long trial-and-error with divine inspiration or knowledge passed from previous generations, it would take much too long to (somewhat) precision craft all the mechanical pieces needed to make it work. Oh, there may be legends of just that happening later on to glorify tinkers and encourage young ones to follow that path, but they'll be exaggerations or entirely made up.
What I could see happening however is a big cultural emphasis on tinkering and teamwork - the two things that let them overcome challenges and thrive rather than just live in constant fear as prey animals. Small nomadic tribes carefully evading predators might meet up in an inhospitable place (where no predators are interested in hunting) every year for a few days, bringing their newest ideas, interesting materials, and if the tribe has been prosperous enough to afford working on those instead of gathering food even pre-fabricated parts. Knowledge would be exchanged, those with the best ideas/materials/parts would be most attractive as mates for partners from other tribes, and the tinkers would work together on projects that particularly inspire them for a bit. After the meet, anything too big to be carried along with a tribe gets moved into a small cave (further excavated a tiny bit every year) that gets sealed until the next meeting with pebbles, mud, beeswax... to protect the half-finished stone gears and axles and whatnot within.
Success: a new age
Finally, after decades of work, the project laid out by the forefathers is completed. A huge (for Lilliputians) pile of discarded gears that broke or were just off the needed size by enough to not work proclaims the generations who went through trial and error to get us this far. Tiny gears carved from rocks or bones in painstaking work, lubricated with fat scavenged from what predators left after sating their hunger. Ropes operated by teams of Lilliputians pulling up counterweights or directly putting tension on the great machine. It takes minutes, maybe even hours - but finally a latch clicks shut. A huge spear (well, a small crossbow bolt) is lifted into place by more pulleys. And the machine that would change Lilliputian history forever fires for the first time, with force once thought to be only available to the large beasts. The spear pierces the bark of the young tree that was used as a target, proving that at least if predators can be forced into the right place (by living in caves with defensible tunnels, for example) warding them off or even killing them will be possible.
Plans are drawn up for the perfect location to set up the first permanent defended Lilliputian settlement - it needs to have access to water, but must not flood. The earth must be fertile, the climate favorable, and there should be a place where defensible tunnels can easily be set up. After much debate a consensus is reached, and the great machine disassembled carefully and put back into storage so it can be transported to said location once all preparations have been made there. It will be years of work still, but finally our peoples' hope is in a workable plan rather than only the dreams of the tinkers.
Bonus: how they got here - surviving as 6-inch nomadic gatherers
This part is fairly easy, I think - at least compared to the rest. Small tribes of Lilliputians forage for berries, scavenge opportunistically, and survive the winter by stockpiling nuts (which would be a pain to break open, but worth it) or mining out root vegetables rather than try to pull them up. Many individuals and maybe even entire tribes will be lost to rats, foxes, birds - but the surviving ones are the ones that learn to recognize tracks of dangerous predators and how to avoid them.
Starting with only 100 individuals as the question states would be hard, a lot of luck would initially be needed. But the stories we hear or read are almost always those of the lucky ones - the hero who finally killed the dragon walked over dozens of charred skeletons on his way through its lair. For every great leader bringing an age of prosperity to their people, dozens more have failed. For every great invention advancing society by leaps and bounds, hundreds of people lived their lives seen as crackpots and died broke without anything to show for their work.
So requiring a little luck for a story is not a bad thing in itself - it only becomes one if the amount of luck is unbelievable, or its influence is so constant that it becomes plot armor. I think our Lilliputians could do it, even if it will take them a lot longer than we might have hoped.