Your son (how old is he?) has several issues that need attention. Best would be to begin with your pediatrician and/or school counselor for advice. The self-induced vomiting may be a behavioral reaction to his class, his ability or inability to learn, his inability to focus his attention, etc. Rarely this might also be related to an underlying physical disorder (for example sometimes metabolic disorders can start manifesting with behavioral problems), which is a good reason to have a medical assessment in conjunction with evaluation for behavioral and learning disorders.
For information on learning disorders, see link below.
Certainly sounds like a serious issue, but not necessarily indicative of a learning disability. I'd find someone competent to do a thorough assessment (behavioral and academic) and learn more from that assessment. It could be that there are serious learning issues which are leading your son to engaging in the vomiting, or that there is some other issue leading to the vomiting. Very rarely does one particular behavior always point to the same cause, so finding a good assessment will be important.
He isn't a stupid kid, but he needs to see a counciler ASAP. Don't be afraid to ask questions like this on websites. He needs to have a talk with a counnciler or talk to the guidence counciler. Homework bores everyone, but that doesn't mean it's impossible to get done. Set aside about a half hour each day for homework. If he gets all of his homework done during school, then just have him read a book. From the sound of things, it seems he is in 2nd or 3rd grade. He might like the Magic Treehouse Series or Horible Harry books. Hope this helps.
I would not guess a learning disability is the first area of concern, although that too should be assessed.
He is desperately making a statement to avoid school it appears.
You need to get to the bottom of what he is feeling in that classroom.
Get him to find more constructive ways to be heard and understood so
things can be corrected and made more comfortable for him to learn
Maybe it is also a peer problem. He might not feel like being amongst his peers, or he is looking at excuses to be home. Maybe try to take him for some professional evaluation. A teacher should partake the role in evaluating whether he has a dissabillity to learning or not. If they have not yet done so, ask his class teacher to evaluate him.