On St, Thomas Admiralty or Blue Island Divers are often recommended, both do a good job and depending on where Disney ports - if It's Crown Point Marina you can walk to Blue Island - if it's the more commonly used WICO dock either will pick you up there, Admiralty with a van, Blue Island with their boat at the end of the pier.
On Tortola you're likely porting in Road Town - Sail Caribbean Divers is nearby("3 hills over"). There's also We B Divin which I believe is walking distance from the cruise pier. IDK much about any of the Tortola shops since we stayed on Virgin Gorda.
The other shops are further east or west and Torrola is a very long island so a cab would be expensive unless they're willing to pick you up. bviscuba.org lists a couple other options.
From Tortola ask about going to the RMS Rhone if you like wreck dving - it's a good one. Across the channel off Salt Island so check to be sure you have enough time in port. They also recently (April 2017) dropped the Kodiak Queen off Virgin Gorda - it's one of few ships to survive Pearl Harbor intact. Painted Walls is a really good reef site close to Tortola - bring a light for the best effect.
