There were a few dive shops in Hoi An when I visited. I don't recall seeing any in Da Nang proper, but I wasn't there for diving and I don't have a personal review. It seems from basic searching most shops list themselves in Hoi An, not Da Nang.
Hoi An, depending on your resort/hotel, is as close as a 20 min free shuttle away, such as from the Hyatt. Or can be quite a bit longer if you're way up at the Intercontinental. Those right in Da Nang city are probably more like 30-45 min away. Anyway Cham Island Diving, for instance, is one operator you might check out their site to become familiar with the area. There aren't many personal reviews here for Vietnam though, so I guess you'll be rolling the dice either way. If you do go, please give some feedback on here afterwards if you can find the time, so we get a few more Vietnam trip reports.