I don't think this is the "norm" per se. Neither of my prom dates ever asked my parents permission to take me to prom.
But I don't think that it's necessarily accurate to say you have no say in the matter whatsoever. I would be nervous about my 16-year-old daughter going to prom with some boy I had never met/didn't know. If you expect her would-be date to ask your permission first, then you need to communicate that to her. A simple, "Whichever boy asks you to prom needs to come speak to me before you accept" would be a clear and simple directive. And if she asks why you just respond, "Because I want to make sure he's good enough for my little girl". She will probably whine.
Another option might be that once she has been asked and has accepted, you could schedule a series of get-togethers with the boy and his parents so you can get to know each other. Or you could combine the two options.
Either way, I would make it clear that you, as Daddy, have the veto option if you don't like her prom-date choice.