You may get so many recommendations on different brands, your head will swim. Everyone has their favorites. You definitely want to laminate, unless it's a short term graphic. If the door has a factory finish, you should be ok, if it's been repainted do an adhesion test since a lot of paints designed for easy cleaning and repelling dirt also repel vinyl. In that case, you may need a vinyl with more aggressive adhesive, but any exterior/ automotive line will work.
Personally I'd go with an air release for easy application, automotive style, something along the lines of 3M IJ-40 polymeric if it's flat, or IJ180 cast if it has texture or profiles that requires something more conformable, top with whatever lam is recommended by the mfgr in the sheen the customer wants. Those are my go-to, but we're a mainly 3M shop.
What brand of vinyl is your go-to? Use what you're familiar with, and can get through your vendors, every brand has compatible lines with everyone else (if it's not 3M it'll probably be lower cost). An air release, either polymeric calendared (so it doesn't shrink), or cast, and appropriate lamination. Very straightforward. Print, lam, cut, clean the door & lay it, collect a check, & go get lunch.