You can do it with vinyl specific spray paint.
If you have a custom colour it might be cheaper to get the paint colour matched and supplied in a tin for use with a spray gun rather than put into pressure pack cans but last time i used VHT cans as the burgundy was good enough for what i wanted.
http://www.vhtpaint.com/products/prepspray/
http://www.vhtpaint.com/products/adhesion/
http://www.vhtpaint.com/products/vinyldye/
A spray gun would probably cover the surface quicker as it has a larger spray pattern and will apply a more uniform layer of paint but the cans work aswell.
The difficulty in spraying used trims is that there is grime and protectant like armour-all (silicone polish etc) which reacts with the paint and it needs to be thoroughly cleaned off (see Vht prep spray) and a primer (see Vht adhesion promoter) can also be used that assists the paint to adhere to the vinyl. I missed this step and the vinyl wore off easier than i wanted to i'd try the primer next time.
New trims should be easier to paint.
If you are able, id give it a go yourself and practice on a used trim first to perfect the prep and painting before doing the good ones.
Also, there are a few places around making repro door trims and im sure one of them could do them in other colours. Might be worth a couple emails to see...