You don't need to use staples. Normal contact adhesive will work fine to hold the edges of the vinyl on.
Here's how I would do it:
Clean the door trim thoroughly.
Paint/spray both the vinyl and the door trim with contact adhesive, getting a fairly thorough coat on each.
Let them both dry to a tack-free (non-sticky) surface.
Start at one end of your door trim and lay the vinyl on from one end of the trim to the other, smoothing as you go. A heat gun makes it easier to stretch it around and into all the corners.
Once you've got the vinyl stuck to the front of the trim, flip it over and paint an inch-wide strip around the edge of the back of the trim with contact adhesive, make sure you get a solid coat on because this is where it counts. Paint a bit more on the edges of the vinyl and let them dry.
Stretch the vinyl around the edge and press it down onto the adhesive.
Neatly trim the excess off and screw it all back together.
I hardly ever use staples when I'm vinyling anything at work, the only time I use them is to pin down wrinkles on the back of things that need a fairly flat surface on the back.
The two most important tips I can give you when vinyling stuff are:
Vinyl is designed to be stretched. Stretch it.
Contact adhesive must be dry for it to stick.