You can paint over vinyl laminate cabinets as long as they are properly prepared and primed. I don't know what "vinyl laminate paint" is. There are a lot of different products on the market. If the cabinet surfaces are properly prepared then any type of paint can be used.
Probably the peeling was caused from the heat. These areas can be re-glued if you can get some glue behind the vinyl. If not they can be cut out and patched. The real key to painting this type of cabinet is cleaning with a strong cleanser, sanding with 150 grit sandpaper and priming with a very good primer. Any areas needing patching can be repaired with Bondo or similar product after priming and spot primed before painting.
This is a doable project that could save you a lot of money. It won't be easy and quick. The real work is in the initial prep work. If this isn't done correctly the paint will peel, ruining your efforts.