I was doing some pencil stuff when my 1950s Stocker & Yale desklamp running two ~1956 GE STD Cool White 4W lamps suddenly turned off. Afterwards I have to hit it a bunch of times really hard for it to work. This has happened a few times and I can't figure out why...

After I hit it really hard a bunch of times, it works just fine again but if I hit it, it will go out again... It seems something it causing a connection to break but I don't know why it would break just 12 minutes after the desklamp has started and it's not even very warm. And its ran for over 12 hours straight without failing...

Any ideas what is wrong with it?

All connections and switches are weak points to verify. It could be a number of reasons, such as a bad wire connection, corrosion on the lamp pins, corrosion in the switch contacts, a weak switch spring, or even some breaks in the ballast windings. If the fixture has starters, it could also be a factor, but starter failures normally only manifest themselves upon starting unless they are stuck.

I wish you all the best in finding the fault(s) and bring this lamp back in business !

12minutes after start sounds like it has been just warming up, so some thermal expansion caused some loose contact to break the circuit. I would start to look mainly at places that get warm (bat that could be the whole fixture, I know)...
You needs to inspect it all for loose connections and oxidized contacts. You will definitely find some...
And better to clean and treat the switches and sockets with some contact cleaner...

Thanks for the responses, there are no starters in it, it's a manual start lamp. I can't check the electrical stuff out myself but I will do it next time I have someone who can help me.

I got the lamp for Christmas a few years ago and I think I'm the first person to use it. The lamp removal process is extremely hard and scratches the sockets up (which previously weren't)

Open the lamp and arrange it so you can reach everywhere with probes without moving it much. Let it run up and go out. In this condition, unplug and mesure what went open circuit thats supposed to have continuity

In a manual start lamp this may be the off switch barely staying close when not touched

Yes, I have a fixture that does this sometimes. I found that the problem was the lamp (light bulb), because whenever I put in a different one it would work just fine.

Hm next time it goes out I will move the lamps around a bit-

Hopefully I don't need to actually mess with the fixture because it's rather hard to open

Sometimes the weird sockets on these don't make good contact. Not sure if that's the problem with yours.

Hm I moved the lamps around a bit while it was running, but it didn't change anything-

Yesterday it ran for over 3 and a half hours without any problems

To me it sounds like a thermal expansion leading a broken connection, likely in the ballast. you said the fixture is from the 50s so i would anyway recommend a full rewire and contact cleaning... These appliances were but to last yes, but they were never intended to be used for 70 years...

I think I got this fixture unused but either way it's old and it's ballast already had some issues-

It makes this annoying high pitch buzzing sound, when you turn it on it actually kind of sounds like a fridge compressor like pitch bend. Someone told me there was like a gap in the ballast's core, which I have no idea what that means...

I think one day I will have a friend help me replace the ballast (with one from another Lite Mite that I don't use) so that will hopefully get rid of both problems...

Here is a video of it... https://youtube.com/watch/nuZQgnIsIR8?feature=share (Turn your volume up)

The gap in the ballast core is an essential design feature, it must be there for the ballast to work.
Unfortunately this gap uses to be responsible for a lot of the field to reach outside of the ballast body, where it is affecting nearby iron/steel structures and makes them vibrate. Mainly when not properly mechanically fixed (lose screws or rivets, inadequate support, too large flat sheets,...)
To reduce or eliminate this field needs some advanced field shaping engineering (without resorting to "brute force" heavy shielding), which came only later (mainly to normal price range products).
And to reduce the noise, the fixture design and construction needs to address the rigidity of the structure mainly around the ballast so it is not allowed to vibrate, mainly eliminating eventual mechanical resonances, or the use of nonmagnetic materials (aluminum, plastic,...).

I think what Emersyn was told about (the mentioned gap in the core) might have been meant not about the intended magnetic path gap, but about a single lamination (the outer leg of the last E in the lamination stack) that broke from the varnishing and is buzzing

In this case, compressing it back to its place mechanically may make the ballast more quiet

Oh, I should try that next time it is open, which will hopefully help it-