|MadSci Network: Chemistry|
You've been reading up on why glue hardens, so you probably know there are different types of glues that harden in different ways (there's an excellent summary in the Wikipedia article on adhesives). Glues work by chemical processes, and chemistry works the same in space as it does on earth. Of course, difficulties might arise when actually trying to apply a glue in space due to weightless conditions, but presuming an astronaut could brush on or somehow apply some Elmer's Glue to his space suit, the water would evaporate and the glue would harden. I should mention that I'm presuming conditions exist that would allow the water in the glue to evaporate (which you may not be able to assume in all cases). If the glue were very cold and applied to something that is also cold, you might have to expose it to the sun to heat up the glue enough that the water can evaporate. Hot adhesives and reactive adhesives should also work in space, but I can think of a couple of reasons why they might not work in all cases.
Things in space typically experience large differences in maximum and minimum temperature. This is due primarily to the lack of an atmosphere that tends to spread heat around relatively quickly. In short, the side facing the sun tends to get quite hot, while the side away from the sun tends to get quite cold. Typically satellites are rotated at a prescribed rate to keep all the parts in a reasonable temperature range, but what is reasonable for electronics and structural components might not be reasonable for a hot adhesive which will lose strength at higher temperatures and become brittle at lower ones.
For another example, consider what could happen if you tried to apply cyanoacrylate (the main ingredient in super glue) to an object in space. It's possible you could run into a problem there that you would never run into on earth: lack of water. Cyanoacrylate hardens due to a chemical reaction with hydroxyl ions that are readily available in water, but there is no guarantee that a material in space will have even trace amounts of water on its surface.
So the short answer to your question is: yes, Elmer's glue (or any drying adhesive) could be made to work just fine in space. However, since there are many different glues with different chemical processes involved, it's impossible to make a broad generalization like "all glues will work fine in space."
Hope this helps,
Try the links in the MadSci Library for more information on Chemistry.