Can you take screen shots during gameplay with GOG?

Yes, so long as you have some form of screenshot-taking software.
Unless you're asking if there's some sort of system ingrained in GOG games where there's some form of automatic function to let you take screenshots. Then no, unless you're using Galaxy.

Print Screen?

If using Windows 8 or higher: The windows button+print screen.
Print screen should be right next to the delete button and written "prt sc". Windows button should be on the left, between the alt key and the control (cntrl) key.

If the game runs under DOSBox, Ctrl-F5 should work, putting the screenshot in the game's "capture" folder.

And now we battle to see who gets chosen as the answer...

Usually it's none or the wrong one XD

*throws puzzle shuriken and pins phaolo to the wall*
No need to have more potential competitors. Two's plenty. I can handle philosophy boy here myself.

I'm gonna take a shot in the dark and guess you're asking about taking screenshots with Galaxy (GOG's optional game client which quite a lot of us don't use) and not GOG.com (the website) itself. ;)
As far as I know, Galaxy doesn't have this functionality yet (though I'm sure it will someday -- bear in mind, it's still beta software); then again, I don't use Galaxy, so what do I know? =D

Hey! Stoppit! How DARE you interrupt my staring contest with fractal dude?! Now I have to figure out how to-
HEY!

I by GoG you mean Galaxy then no. Not yet anyway.
This will come later on.

Add it as a non-steam game to the Steam Launcher, and use F12 for screenshots :P me will :P

Get MSI Afterburner.

Windows 8/10's screenshot button works wonders when it does work. Some games don't even let it do its magic, and end you up capturing a black screen.

Not only does MSI Afterburner allow you to take screenshots, it can record video, check your framerates, you know, cool stuff done better than what Valve could dream of.

Lol.
Btw, do you know if there's a free program that auto-saves and auto-numbers screenshots?
Hypersnap worked quite good for many games, but it's commercial.