To bend plastic laminate you would need to heat it with a heat gun being careful not to burn the plastic. Even with good heating techniques there still is a limit as to how much you can bend the stuff. The minimum radius that you can bend laminate is measured in inches not fractions of an inch so if you are wanting to bend it over a rounded edge it is not possible. I do know that there is vinyl sold in rolls but I don't think that it is in big enough widths to cover your cabinet sides without leaving a seam.
Vinyl is sold in rolls that are wide enough (e.g. 3-feet wide), but it may be difficult to find. Different laminates have different abilities to bend. Someone at a home center (Home Depot, Lowe's) might know more... though I wouldn't bet on it.
Paint would be your best bet in this case. Make sure you fill in any holes or imperfections in the plywood. Then take and sand the bare wood starting with a 80 grit paper and working your way up to 200 grit you could get a real smooth base to start with. Then paint with 2 coats of primmer followed by 2 coats of paint and finally with 2 coats of clear. Make sure that you wet sand between each coat with a minimum of 400 grit sandpaper and wipe off all sanding dust. Taking the time needed you will end up with a nice glossy or satin finish depending on the final paint you use, without the wood grain showing.
I would offer slightly different advice about painting... If you start with a decent quality plywood (e.g. furniture-grade rather than CDX), I'd start sanding at 150 grit and stop at 220. The surface veneers on most plywood are very thin, and relatively smooth, which is why I wouldn't go all the way to 80 grit at the start.
I'd put on a coat of primer and see how it turned out. If it seem thin or soaks into the wood a lot, I'd add a second coat. If the primer covers well, you may not need a second coat. After that, I'd follow the manufacturer's recommendations as far as painting and sanding. If you use a house paint and a roller with very little nap, you may not need to sand between coats.
Finally, I probably wouldn't bother to clearcoat the project. I'd buy a paint with the surface sheen I wanted (probably matte, but maybe semi-gloss) and just leave it at that. If you're worried about cleaning, semi-gloss is the way to go. Or try Benjamin Moore's Regal Matte: it has a fairly flat sheet, but contains additives that make it easier to clean. I've been fairly impressed with it...
For best results, it's always a good idea to paint a piece of scrap lumber using exactly the techniques you're planning to use on your cabinet. If it doesn't turn out quite right, you can throw the scrap away and try something slightly different. If you start painting your cabinet and find out that it's not working... that's obviously a
much bigger problem! Get everything right on the scraps, then move to the cab.