Hi. Welcome to cruise chat. My husband, 2 teenage boys and myself did this a couple summers ago. We were on Princess, in Ketchikan in July. We had a BLAST.
However, I looked at the video on the Princess excursion website for the kart adventure. They go to a lake it says, and we went out to a river/stream with a waterfall that fed it. Maybe you can't go there anymore, the road was a bit tricky, and some karts got stuck for awhile, but managed to make it out. The give you raingear to protect your clothes and shoes and helmets. What made this really fun, is that it had rained, and the roads were VERY muddy, and it was still light raining. It truly was a blast. We did not get dirty, with the gear we had on. We saw a few bears too. The ride back on the boat, was also a blast. It is a VERY bumpy tour, just like it says. We signed up at home, prior to leaving, because we really wanted to do it. We didn't want to risk it being full, when signing up on board. I would definately sign up for this via the cruise line. You really go out to backcountry. If something happens, they will notify the ship, and they have to wait for you. A good example, we were in Belize, drove about 1 hour to a river, took a 1 hour speedboat ride down the river. Well, one of the boats broke down. There we are in the middle of the Belize jungle in a broken down boat.
Luckily we were all booked via the RCCL shore excursion program, and they finally got the boat started. We get going, and narrowly made it back in time. So, as a rule of thumb for me, if we are going to be far from the ship, I always book via the ship shore excursions. If it is a city tour, then it would be safer to book privately with the tour company. Have fun! Lil' Lori