I couldn't find anything about how common vets do it, if it is inclued in vet school or any owners experiences who asked for one.
Are a lot of vets aware of this procedure?
Is it common for them to hear about it?
Is vasectomy accepted when you must nuter before adopting in a shelter? (if not already done)
It's not that commonly done, currently most people will neuter their dog or leave them intact. There are some papers coming out about the risks and benefits of neutering, and more people are inquiring about alternatives, so I'm guessing we'll be seeing more vasectomized dogs in the future, and more vets with experience with the procedure.
Vets are supposed to know enough about vasectomy to have a helpful conversation about it. Many vets might have little to no experience with performing the procedure, or managing complications associated to it should they arise, because it's not that common a procedure. If they can't answer your questions, or if they can't offer the service, they should at least be able to refer you somewhere you might get it.
It is more and more common for people to inquire about it, mostly because of the recent controversies surrounding health risk/benefits associated with desexing.
as for the last point and:
the purpose of neutering for rescues is to get them sterile
The purpose is partly to reduce unwanted litters, but also to reduce pet relinquishment or 'adoption fails', which nobody ever talks about. Vasectomy would have no effect on the later, which is why I think it would be unlikely to find a shelter that goes with vasectomy.
Wondering what a vet would think about a request to render the pet sterile while maintaining his ability to copulate?
That's probably why it's not the most common practice. Most responsible owners feel they can prevent unwanted litters without putting their dog through surgery and leave them intact, or don't feel like dealing with an intact dog and want them neutered. The case for vasectomy is often people who want to be extra certain they don't cause an accidental litter, people who read about it online and are curious, or people who are contractually required to sterilize the dog (for example by breeders, landlords, local legislation, etc) but don't want to remove the hormones.
That being said, if you're insinuating asking about vasectomy it would raise flags, the dog can copulate just fine without having a vasectomy, I think the vet would just think you're exploring all the options for what's best for your doggo.