I've always wanted to play Fallout, so I recently bought White Label's Fallout Collection however, every time I play Fallout, I get a black-screen, which reveals itself when any movement occurs on the screen.
I'm worried if I purchase the GoG's version of Fallout, I will get the same thing.

it happens on the vanilla, but not as you're describing it. the patches listed in the mods section work just fine.

I doubt it will happen in the GOG version, GOG modifies the game to be more compatible with XP and Vista

I had the same problem a few years ago on my old system, I had bought the dual cds and tried to install both games.
FO 1 refused to work (was using XP at the time) but FO 2 worked fine. I have recently bought both via GOG and tested FO 1 so far, and it works fine.
GOG basically promises that the games will work with XP and Vista, and you have to keep in mind even setting FO1 to Win 95 mode will not work if you have a copy of the original game.
You may be able to find a mod to patch it and get it to work, but I never had any success. I accidently threw both games away about a year and a half ago because they were in the wrong jewel. I'm glad GOG has them!