I have had a lot of experience dealing with this problem, and here are some suggestions of things I've discovered over the past few months dealing with worsening clinical anxiety myself.
Adjusting anxiety medication is tricky business, and can take weeks to get it right. Most of these suggestions are things to try in this situation, where you may not have another person to go onstage, and where there are only a few shows left in the run. Most of them are not ideal, but I'm giving them under the impression that you need to try anything and everything you can think of to get this guy onstage.
Is the actor on as-needed anxiety drugs (xanax, clonapan) as well as long-term drugs (zoloft, lexapro)? If not, he should consider getting a prescription for something that he can take right before the show. If he is on them and they don't help, have him check with his doctor to see if he can take a larger dose. 8 more shows of bigger doses shouldn't hurt him too much in terms of building up tolerance, and if it gets him through the show then that's a good thing. These drugs might make him sleepy, but that's better than throwing up. He also might want to look into anti-nausea medication - compazine and reglan are both good ones (he needs a prescription).
Work with him to plan out what to do if he starts to feel nauseous or thinks he's going to throw up. In my experience, it is MUCH easier to deal with anxeity-related nausea if you have an exit plan, so to speak. sievep's suggestion of hiding trashcans onstage or having them immediately offstage is excellent. If the actor knows exactly what to do when the major anxiety hits, it will help him cope with it.
Does he have a loved one who could be with him offstage? Having a person there who he can sit with and receive encouragement from could be extremely helpful. Personally, spending time with my mom always helps to deal with anxiety. Does he have someone similar whose presence could help to calm him down?
Chewing gum helps release a lot of tension and also relieves nausea. Again, not ideal to send an actor onstage chewing gum, but it's worth a shot if nothing else is working. Spearmint is best.
Can he adjust what he eats in the several hours before the show? Another not-ideal solution of going onstage hungry, but it might work if he's willing to try it.
I have a recording of the Dalai Lama doing a healing chant that I've found extremely helpful when I need to calm down. If you like I can email it to you. The track is about an hour and a half long.
Is he seeing a therapist regularly? See if he can schedule sessions right before each show. If he's able to talk and release pent-up anxiety, he may not feel so anxious going onstage.
Finally, have him go see his doctor ASAP (either psychiatrist or general practitioner). The doctor will be able to assess whether or not he should be onstage. If the doctor tells him that he shouldn't be in this situation right now, he should take that advice.
I sympathize with this actor. It's so hard to be dealing with such intense anxiety that you're unable to work. If you have someone else who can do the role, I would recommend that you discuss it with the director and the actor who is having trouble - it will be upsetting for the actor but healthier for him in the long run, and also better for the rest of the cast. I agree with sievep that the rest of the cast needs to be considered too.
I hope that you and your cast and crew are not getting angry at the actor - as Matthew said, anxiety is a serious medical problem, and the actor can't help what's happening to him, and probably feels guilty and ashamed. PurrKitten is absolutely right that he's feeling anxious about being anxious - it might be worsening his symptoms. If everyone deals with him with understanding and kindness, it will help him a lot.
