Get a second opinion from another Vet (or however many it takes to get to the root cause).
The feeding suggestions given by SL are excellent but they at least know what is wrong with their cat.
It’s obvious that something is seriously amiss here and I wouldn’t rest until I knew what.
Once it’s diagnosed, there may be a treatment for the cause.
I had a cat with the same symptoms and the first blood test picked up nothing abnormal except hyperthyroid.
But she was already more listless than usual so the thyroid meds did nothing other than make her more so.
Several weeks later (when she was already starting to look yellowish) a repeat blood test showed her liver to be totally out of whack.
But by then there was little to be done. By the time most cats show any signs of illnes at all, they are usually VERY VERY ill.
It is instinctive for them to hide any signs of illness or weakness and they’re very good at it.
I would urge you to try another vet until you find one who can get to the root of the problem.
The vomiting is just the symptom. You need to find out the cause of the symptom or else things will just keep going downhill.
If there’s an intestinal blockage or infection it has to be dealt with. There are so many possibilities here. Find a more curious vet. I wish that’s what I had done.