hehe We've been to the Caymans. For the life of me I can't remember how we got from the airport to the hotel. Probably a car was sent for us. You can go across the island in about a half hour from nearly tip to tip. It's not a big place. We rented a car and drove around a bit and took ourselves to the airport on the last day.
They do have some American cars, which is weird (left hand drive) but when we rented we rented a right hand drive car. It was straight from Japan, for the Japanese market, so someone had taped a kph to mph conversion on the car. I had a horrible time getting Piper's Radian in. It was a 2004 Civic, but no LATCH or tethers. So I wouldn't bank on having anything more than a seatbelt. That would eliminate the Safeguard Go, the travel vests, EZ on, and the Radian may be a bear. You may get an American car, but if it's a local taxi then they may be driving a RHD car, since it'd be easier for them as they drive on the left.
What are they sitting in on the plane? And how much do they weigh? There are some backless boosters that go down to 30 pounds (we have the Cosco High Rise, $15), and if they're pretty good sitters I'd probably be ok with them in those for the short rides with you guys right next to them. There's a lot of traffic heading up from Georgetown (where are you staying?), but it's on two lane roads, so it's just congestion. There's only a few miles of four lane highways.
Or you can call the hotel and see if they have a shuttle service. When we've gone to Jamaica, Costa Rica, and I believe the Caymans they always had a shuttle and it was a regular car. I can't remember the Caymans (sad, huh?) but in Jamaica it was a Lincoln Towncar (only car a LATCH install with the Radian was much much worse than the seatbelt), and in Costa Rica it was a Ford Expedition. The drivers simply waited while I installed Piper's seat. No meter running, no nothing.
HTH
Wendy