Thanks for all the replies guys. They asked for the credit card number to authorize $100 to confirm my booking. I have a firm booking with them for the first day of diving only, which is a local 2-tank dive. They couldn't confirm Blue Holes or Turneffe as they weren't sure if they were going to have either going out while I'm there, and promised to try to work with other dive shops if needed to get me on the boat (I guess they know who goes where when).
My guess is they just want to confirm I'm not a flake as I never dove with them before, and they are probably busy enough to where they could potentially fill up for the day. They are the highest rated dive shop on the island on Trip Advisor and their rates are the lowest. So far I'd definitely prefer to dive with them. Their cancellation policy isn't anywhere on their web site and what Emeliano said in the email is a bit confusing - "To confirm your booking, we ask for you to fill out the attached credit card authorization form in the amount of $100. We do not run any charges until the day of your activity or in the event of a no show or last minute cancellation, including illness. We reschedule or cancell without cancellation fees if we have inclement weather. ". I guess you have to cancel the day before, or else they charge you for whatever you scheduled, including in case of illness.
I'm not planning to cancel or no show, and I guess I would know if my flight got delayed or cancelled to the point where I'm a day late arriving on the island.
Thinking about it, giving them my credit card number is no different than handing my credit card to a waiter at a restaurant who walks away with it and does god knows what. All things considered I'd trust that dive shop more than most other people considering overwhelmingly positive feedback on Trip Advisor, Google, and this board. I'm just being paranoid based on the number of times my credit card number was stolen somehow.