I've used both marine and Sande plywood in my boxes so I tested both. The marine plywood has fairly hard exterior plys whereas the Sande is pretty soft. I tried four variations, plain sanded plywood, polycrylic treated, water-based primer and an oil-based primer. Each panel had one side that included the 3M tape primer for a total of 16 variations.
After sanding all the pieces, I applied my different primers, let those dry for several hours, then sanded them smooth. The polycrylic was the hardest to get smooth, it tended to pill up under the sander. I wiped them all with tack cloth and applied the vinyl. I didn't bother trying to overlap it, I wanted to have exposed edges so I can see how it adheres. I'm going to let them sit for another day before I see how they stick.
So far the best appearing result has been the marine plywood with oil primer and no 3M. I used black matte finish because I figured that would readily show flaws. Assuming some of these variations stick well enough I'm going to try some other patterned vinyls that might be more forgiving of small surface irregularities.
I'll post an update after I see what sticks.