How do you paint over vinyl-looking kitchen cabinets?

Hi Milly, you should have a problem painting over your cabinets no matter what the material as long as you clean them properly (with a degreaser or natural TSP) and then use the correct paint. For something that is not natural wood I would prime them first and then use a good quality paint. There are a lot of good ones out there now that are great on cabinets. Here are some great tips too...

https://www.hometalk.com/diy/kitchen-dining/kitchen-cabinets/tips-for-painting-kitchen-cabinets-35495116

Clean with TSP and then prime with oil based primer and paint.

Remove all hardware, doors and drawers . Mark on back where they go.

How to Paint Cabinets

Make sure they are clean and dry. Remove the doors and hardware. Mark the doors and cabinets with tape where they go. Lightly sand the doors and cabinets to remove any gloss and roughen the surface for paint with 220 grit sandpaper or a green Scotch Brite pad.. Use a tack cloth or damp rag to remove dust after sanding. Prime with a stain blocking primer like KILZ. Acrylic or water base paints are low-fume and clean up easily with water. Alkyd or oil-based paints require good ventilation because the paint contains solvents that can irritate your lungs and make you feel sick. Alkyd options require mineral spirits for cleanup, but they provide a hard, durable paint finish. Whichever you use, buy the best-quality paint you can afford for a lasting kitchen cabinet finish. Seal with at least three coats with a water based polyurethane. Use a small foam roller and foam brush for a smooth finish.

Milly, a sprayer would likely work so long as you really cover well the surrounding area that you don't want to paint. I doubt you would save much time though.

William's instructions are great!

Sure can! You would get a better and smoother finish. A lot of people would not want to spray paint in the home. Just make sure you cover everything you don't want to get overspray on. Floors, walls, etc.