I would say, in most day-to-day situations, we simply "hold" a sandwich, or "pick up" a sandwich. These words you're asking about are a bit colorful, and they normally imply some kind of extraordinarily strong grip.
"Snatch" means to grab something quickly, and perhaps steathily. I might say:
The homeless boy snatched a sandwich off the tray, and stuffed it into his pocket.
"Grip" means to hold something in a way that would make it hard for something to come lose. Bread is generally too soft for a verb like grip, but I suppose I might say:
Having his first meal in three days, the prisoner gripped his sandwich greedily as he ate it.
"Grab" is unusual, in that it has an informal definition, which NOAD defines as:
informal obtain or get (something) quickly or opportunistically
So, I could see myself saying:
I'm kind of hungry; I'll grab a sandwich before I come over tonight.
but, in that case, I'm not alluding to how I'm holding the sandwich – only that I'm going to get something that is quick and convenient. It implies I may buy the sandwich somewhere, or, if I'm making it myself, I might not put as much care into it as I would if I wanted to make something fancier or more delicious (for example, maybe I'll skip the mustard and just put the meat on plain bread).