I'm gonna start a gaming youtube / twitch channel soon and I'm planning to cover older games availivle on gog - not exclusivly, for instance I will also take a deeper look on Smite - but ich have over 200 games here on gog while less than 10 on steam.
So while I'm a pretty poor guy, setting up the studio has cost me some money and my pets got sick last month, which cost me even more. All that's left now is to buy some new clothing, but I'm already in dept, which brought me to the idea of just requesting t-shirts from you. The offer is, you provide me with gog shirts, and I will wear them on my videos. Free shirts for me, free advertisement for you.

I was thinking of doing something similar, but my upload speed is so abysmal (70k/sec) the resolution would be crappy... Unless it's something that doesn't do a lot of screen changing (like FTL) in which higher resolution captures still look good merely because it doesn't need the bandwidth. I tried with Sacred Gold, and it looked horrible...

You'll have better luck if you already have a channel established. Many companies are willing to do these kinds of things - but not willynilly, otherwise they're really just giving away free stuff to random people on the internet.

Edit: and by established, I'm not even on about x amount of views or having made a name for yourself, just the content.

If you want to do some advertisement for GOG you could just say where you got the games from and put a link to the gamepage into the vid's description. I don't think you need t-shirts for that.
Also, most people want to see the games you play and not necessarily your chest, t-shirt or not.

well, that does depend... O___o