Time to answer some more questions and make some corrections,

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We are staying at the FQ in June. I know that you rent the Sea Raycers at the POR. Where are you allowed to take them? Do you need to make reservations way in advance?
As I said before the entire canal system that connects Port Orleans, DTD, Saratoga, and Old Key West is fair game.
If you rent SeaRaycers on Seven Seas Lagoon and/or Bay Lake, Please stay well away from the larger WDW Watercraft (launches, cruisers and ferry boats) operating on these 2 lakes. WDW Watercraft pilots have no resevations about reporting those operating SeaRaycers unsafely to Lake Patrol who will then take appropriate action to include terminating the unsafe operator's experience.

+1
DS(13) wants to do this in a bad way and I inquired how much just yesterday. I was told $34.12 for a half hour including tax. We'd be renting from YC cause that's where we're staying.
This is correct, the prices I listed earlier were pre-tax.
Are you allowed to switch drivers halfway through the ride? For example, my son and husband go out for 45 minutes, then come to shore/dock and I get on with son and go out for 45 minutes. Or would the swap out take too long if you have to do another set of paperwork, etc?
Most locations will allow you to do this. You will need to return to the dock you rented from in order to switch out drivers and/or passengers. All of the drivers and passengers under one rental name can be done on the same paperwork. This is easy to do on the smaller lakes on property, but keep in mind that Bay Lake & Seven Seas are very large so it takes time to return to your home marina.
Are the boston whalers available at all locations? Mainly bay lake and seven?
They are available at all locations, but just like the Sea Raycers their speed is limited on smaller lakes.
Is it like a gas and no gas pedal? Steering wheel like a car?
All of the boats use a typical steering wheel like you would find in a car, however the steering does respond a little bit differently but does not take long to get used to. None of the boats use gas pedals, they are all controlled by a
throttle system. This might scare people, but there are no brakes on boats, you must shift into reverse to completely stop your forward motion. Again, this does not take long to get used to and the lakes are very open.
How easy is it to find available boats during June / July? Do they all get booked up quickly or are boats usually available? Is this something I should do in the morning in order to find an available boat?
Most marinas have at least 15 Sea Raycers on hand, the marinas at the larger lakes have even more. If for some reason all of the boats are out, it typically does not take long for some to come in. Morning is definitely the best time to rent a boat, but also be aware that many people think the same way and get to the marina as soon as it is open. But don't worry, it doesn't take long for boats to return.
Also ... I haven't been to Florida during the summer in many years, but as I recall it was a daily occurrence to have afternoon rainstorms (usually around 3:00). Should I expect this in June? Are there better / worse times of the day to take the boats out?
From late May until early September the 3:00pm thunderstorms are like clockwork, it's amazing. Take a boat out in the morning to enjoy a rain and lightning free experience.