Yes, it's a funny question - which we've all been thinking about for a long time.
Most spiders inject their prey with a combination of chemicals that immobilize the prey (dinner) and then begin to melt the prey into what they consider the edible parts. After they suck up the liquid parts of the dinner, only bits are left, but no, they don't throw up. Spiders, if their webs are disturbed, will 're-absorb' the part of the webs that they need to remake because for a spider, it takes a huge amount of energy to generate webs to make a web. The use of spinning nozzles, and some of its 8 "hands" may look like it's "throwing up" but that's not what's going on... ...so you know :-)
So the answer is: No, Spiders don't vomit :-)
Dear Katja N-Masters,
Pet sitter asst.