I'm a pediatrician who used to do ear piercings. We don't anymore. Funny as it may seem, my recommendation for where to get ears pierced is not what most people expect. I tell people to go to one of the piercing parlors (piercing pagoda, etc...)!
There used to be concerns about cleanliness, but now that the equipment is basically all single use (at least everything that comes in contact with the child), the infection concern is minimal.
While I am of course pro-immunization, and I tell my patients who are getting immunizations to at least wait until after the first set, I would tell my patients who aren't immunized to wait a little longer (preferably until at least 6 months), and then just simply watch for signs of infection.
Again, my concern isn't infection because the equipment isn't clean, but simply that, until it heals, it is an open wound.
Why don't I do it? They do a lot more of them than I ever did, so, they're faster, it's less painful, and they're better at symmetry! Of all the things I need a mother complaining to me about, asymmetrical piercings isn't one of them!
You shouldn't have too much trouble finding a place that will do it on an unimmunized child. You might have to make a notation on the waiver, but it shouldn't be a big deal.