Just did it for a 3rd time back in November. Booking - as far as I can tell you can only do so by calling them. It's not something that a travel agent can do for you.
You can bring your own mask, and dive watch (I've taken my wrist computer with me each time, but that's borderline). They provide everything else. (Bring your own swim suit though) If for some reason you absolutely can't use their provided shorty wetsuit, apparently you can supply your own but it has to be BRAND new (as in, tags still on it). The gear they provide is decent, and they appear to have a range to fit pretty much anyone (supposedly wetsuits up to 6x).
The Atomic regs are configured with a single 2nd stage (no octo) and a pressure gauge. The DM's in the tank are your alternate air source if for some reason you couldn't make it the 20' max to the surface. They use AL63 tanks, which for the depth and planned dive time (45 minutes) are usually plenty. If you're an air hog though they will change you out if there's enough time left.
If you have family with you, bring them along to the meeting place and they'll get a briefing on when and where to see you. I think they also got to come back stage with us afterwards. The video is US$35. You also get a little drawstring bag as a souvenir. (They used to do t-shirts). The whole thing is right about 3 hours. 1st hour is tour and gearing up, 2nd is parading thru the exhibit to the tank, and doing the dive (this is when family can see you), and last hour is showering and watching the video. Every time we've done it we've also been let out into the park as well.
There's lots of examples of the souvenir video on YouTube. First part is boilerplate about the program (like how the funds go towards conservation) then the actual dive footage. If you have folks there to watch you, the'll make a point of getting footage of them and you thru the glass.
Best part of the dive is being "on stage" and interacting with guests. Swim up to the glass and interact with a kid and you will blow their mind.
