Yes you can paint those vinyl walls
I searched so many times to get tips on painting the vinyl covered walls that I gave up and did it. Its been almost a year now and no issues so I thought I would post here for those who have the same fears.
For the bathroom walls (slick surface, no texture) I used a good oil base primer. I also used an oil base paint. I have had not peeling issues what so ever.
For the textured vinyl I used regular Walmart primer and Walmart paint. No issues at all!
Kitchen, regular primer and semi gloss paint from Walmart.
For the ceiling, I got rather tired of primer and just used ceiling paint. No issues.
I will say that we smoke, and all walls were thoroughly cleaned with plain dish-soap and water. I would do this even without the smoking.