This is not generally done, although I'm sure I have seen the vice being used for glueing up certain jobs, mostly between the jaws, I can't remember the sources. One passing reference to this the poster said he'd never do it again though, since the glue dripped onto the vice screw and guide rods and made a mess that was tough to clean up.
Too much strain on the vice
The main reason not to rely on this is because a vice isn't capable of exerting the same pressure as clamps can, at least not without putting excessive strain on the mechanism.
You can get away with a light hold using hide glue (or no clamping pressure at all in the case of a rubbed joint) so if you used hide glue much then this could be a practical way to do some smaller glue-ups.
But since most of us are using PVA glues primarily these days (both white and yellow types) which rely on very high pressures to ensure the strongest bond you'd really need to crank the vice down tight and you'd risk wearing it out or breaking it — they ain't made like they used to be!
Ties up the vice
Another reason that occurred to me is simply that while the thing is drying the vice is tied up and you can't use it, and that area of the workbench, to work on something else. Even in a home shop this could prove to be a bottleneck to production.