I am sorry to hear about the problems Max is having. Before we go onto prednisone I think some more testing sounds in order. If your vet is not interested in further diagnostics, then we need to find someone who is. Prednisone is a steroid and should not be used lightly. Also FYI Prednisolone (notice the extra syllable) is generally accepted as being safer and better for cats. Prednisone is an inactive form, prednisolone is the active form. Dogs and people can easily tranform prednisone to an active form, cats have a much harder time. One of the problems with steroids at this juncture is the fact that it may mask a more serious problem and can make certain problems such as chronic pancreatitis, infection or foreign body diseases worse. Disorders such as inflammatory bowel diseases are common in cats and are readily responsive to steroids. However food allergies will also be and can be more easily treated.
So what diagnostics can be done?
1) Fecal parasite screening by centrifuge to rule out parasites.
2) Empirical deworming may be beneficial regardless of test results.
3) Radiographs to look for organ enlargement, kidney stones, bladder stones (yes, these could cause recurrent vomiting), foreign bodies, masses, etc...
4) Urinalysis and culture. I have had several patients, both dogs and cats which have urinary infections and have presented with on and off vomiting.
5) Food trial - a hypoallergenic diet (not available over the counter) needs to be used for a minimum 8 week trial. The antibodies can remain in the gut for at least 8 weeks so it takes a while to know if a hypoallergenic diet is working. These are available either by prescription or home made diets (I am sure there are plenty of people here in the forum to help offer hypoallergenic receipies).
6) Additional blood testing such as in depth viral screens and a feline specific PLI test to look for a chronically inflamed pancreas.
7) If these either do not solve the clinical signs or lead to an answer, then I would recommend an ultrasound with an internal medicine specialist.
There are other diagnostics available but they can be recommended as your veterinarian and you get further involved into the case. Each patient's workup will be slightly unique.
I hope this helps! If you are looking for some options without going to a veterinarian, then I would suggest over the counter deworming, a food trial for 8-12 weeks and perhaps a trying pepcid (famotidine) - your vet can give you an exact dose based on weight.