You can use vinyl paint - but only if it is vinyl - it will peel off of anything else.
Your best bet is to get krylon plastic paint and primer - and spray it - yes, I know you don't want spray, but that gets you the best look - factory look. Otherwise you will get splotches from stains or brush marks from liquids.
It helps to know the material - and then get the right paint. In a high use area you will want to follow the directions to the T and do several coats - letting it cure between and after coating.
It has to have a special base to adhere to the plastic. The same paint they use for bumpers - so if you have ever shot a newer car you get both mixes -- the krylon stuff is cool though, it really looks like the plastic is colored and not painted. I haven't tried shooting bumper paint on a textured door panel but it should work - just get the coats real thin.
PPG should have it - no problem. Just get the right solvent to thin it and try it on the back first!