Before silicon computer we used the vacuum tube in place of transistor, the computers then were extremely bulky and took up a lot of spaces. Would it be possible for a team of researchers and engineers to land a man on the moon in the first half of 20th century using mainly vacuum tube technology? Goal is to replicate Neil Armstrong's feat using vacuum tube technology only, handwave cost and fuel.
The main problems with vacuum tube technology is that they are bulky, heavy and require a LOT of power to run. Most of that power turns into waste heat which needs to be dissipated, adding to the problems.
You can avoid all these problems by locating the majority of the vacuum tubes on the ground instead of in the spacecraft. You would replace the AGC, GNC and AGS with decent radio communications, remote reading of the sensors, and control of the spacecraft equipment.
There would be a time delay too, of course, but that is a challenge more than a show-stopper... the soviets drove moon rovers around for months by remote control for example. The early soviet Vostok and Voskhod spacecraft used vacuum tube technology. Early space probes used Vidicons (a type of vacuum tube/crt) for imaging, so a lot has already been achieved with the technology.
Obviously the development of integrated circuits was a major breakthrough and NASA pushed this direction hard, but with the resources at their disposal back then, I still think it is plausible that a remote controlled manned landing could have been performed, had they really needed to do so. NASA had a strong design philosophy of having the astronauts in the control loop, and being able to fly the mission even if all communications were lost, but this is not essential to performing a landing.
While not limited to vacuum-tubes only, Soviet spacecraft used vacuum tubes for certain subsystems, and continued to use them for several decades after the Americans had switched to all-silicon components. They did land on the moon -- see the two Lunokhods. The Russian reasoning was that it was much easier to make vacuum tubes radiation-proof than solid-state devices. Sometimes this reasoning made for interesting results, see the Zond 1 "mission".
The question being about the problems of early space technology, the following quotation may be of interest:
The many problems that Gagarin faced on his mission were not necessarily due to poor design or bad engineering, I would argue, but instead a combination of haste and poor workmanship on the factory floor. Consider that the Vostok spacecraft consisted of 241 vacuum tubes, more than 6,000 transistors, 56 electric motors, and about 800 relays and switches connected by about 15 kilometers of cable. In addition, there were 880 plug connectors, each (on average) having 850 contact points. A total of 123 organizations, including 36 factories, contributed parts to the entire Vostok system. Despite redundancy in a large number of systems, human-rating such a spacecraft with absolute confidence was practically impossible. Yet, the way that Soviet engineers designed the system, it was meant to operate even at the blurry edges where parameters were pushed to the max. It is because of this that I would argue that the Vostok design was in fact excellent engineering if we define “excellent engineering” as also being incredibly robust. ("Declassified documents offer a new perspective on Yuri Gagarin’s flight" by Asif Siddiqi, in Space Review, 2015)
It is absolutely possible to use vacuum tube technology to get to the Moon, and it is likely possible even using mechanical computers as well.
Highly sophisticated mechanical computers have existed since antiquity (see the Antikythera mechanism, reaching their apogee with ballistic fire control computers on WWII era battleships. Indeed, it may be more viable to make a mechanical computer rather than a vacuum tube computer so long as you make the computer for a very specific set of tasks.
Vacuum tubes computers have many disadvantages (as noted in other answers), but the key advantage of electronic computers is they are easily programmable to do a much wider variety of tasks. A rocketpunk spaceship with a vacuum tube computer only needs one device, whereas a steampunk spaceship using mechanical computers would need many different devices, or a rather massive singular mechanical calculator (see the Analytic Engine) to do the same things.
Realistic spacecraft designs and plans have been published since the 1930s by such organizations as the British Interplanetary Society or Verein for Raumschiffahrt which obviously were based on the idea of either unassisted human spaceflight or using very minimalistic computer assistance (think of the scene in Apollo 13 where Ground Control pulls out a slide rule to make a quick calculation). Werner Von Braun wrote a fully realized plan for a Mars mission in the late 1940's and early 1950's, which was probably doable with the technology of the day.
For the most part, the rules of spaceflight are controlled by a small set of equations (the Rocket Equations) and ballistic equations. Much of what is needed for space flight (once you launch) could be replicated by the sorts of skills navigators have been doing since Elizabethan times using a sextant, and calculations for doing the retro burn for landing on the moon and taking off again are conceivably doable by a tailored mechanical computer.
No. You can find presentations on youtube etc. on the history of computing. Computer guidance and controll systems were enabling to get launch vehicles to work.
Even as project Apollo was underway, digital logic was getting smaller and lighter.
But with some handwaving…
To try and come up with an alternate history, you could look up the requirements of the earliest systems, and use that as a minimal amount of computing power. See how heavy that would be using tubes. Figure mechanical solutions for navigation etc: how bulky would the instruments be?
If a single launch vehicle can't carry the needed mission, assemble a lunar transit vehicle in orbit using multiple launches.