I've rented in Cozumel on a cruise, and in St. Lucia on a non-cruise vacation, both with
Avis. I think they were manual in both cases (definitely in Cozumel), but you can probably reserve an automatic if you prefer.
It's worth keeping in mind that the local companies are franchises of the large rental agencies, and they do not necessarily use exactly the same procedures or have quite the same standards as you'd find in North America. That's true even in Europe (they're not necessarily worse, just a bit different).
When we returned from Cozumel I had to have my credit card replaced due to fraudulent charges in Mexico. I know it was the car rental place, because I used that particular credit card exactly once on the entire trip. I don't recall the card being removed from my sight, but it's not difficult to memorize a 3-digit CCV number for the five or ten minutes they would need before writing it down.
I didn't have any trouble in St. Lucia, but the service was sooo slow. IIRC the agent was on a break or something at the time the car was reserved, and then it just took forever to do the paperwork.
Also make sure you check the hours. Often the offices close early, and sometimes also have a mid-day break, and there may not be anywhere to just leave the car when you're done with it. In Cozumel we had to park on the road several blocks away from the port. We went back to the ship briefly, and then I went back out to collect the car and return it after the office re-opened.
Having said all that, I would for sure do it again. I find renting to be both far cheaper for a family of four and more flexible than other transportation options, so we do it frequently. But it's good to know what to expect.
