OK, I solved it. First, I called the closest dealer, and they were very nice. They said I could just stop by and they'd be happy to install the plates for me. That's really inconvenient for me, so I asked about the screws themselves and they said the front plate takes M6 screws, 10 or 12mm long. I did not realize the bracket was a separate piece - I thought the bumper just had a flat spot for the license molded into it - so on closer inspection I decided to remove the bracket and take it with me to the hardware store. Good thing I did. The bracket is plastic and has holes molded into it for the screws, but no threads. I'm sure M6 screws would work just fine, but whoever first put a plate on this car used 6-32 screws, so that's what I bought. The screw heads weren't very large, so I bought washers as well, all in stainless steel. Probably the only non-metric fasteners in the whole car.
So for future reference, if anyone ever searches here, the front plate (on an E320 at least) is held on with M6 screws but 6-32 will do in a pinch, and it's more important to go with the threads already cut in the plastic than to cross-thread it by using some other screw.