Well my baby is 13 but here goes my advice anyways!!
We stay at the Surf Club and the pool has an area that has a round small wading pool that might be a nice place to sit with your kids. Seven months olds don't do all that much but enough that sitting in a stroller all day isn't that much fun. That area has a fairly nice shady area. I am not sure about how busy it is in May but I would imagine it must be easier to get chairs, umbrellas etc at that time.
As far as a car goes, I would say rent one! I was always very particular about putting my babies/kids in car seats so if you rent one then you can always have the car seat available. I believe many of the car rental companies have car seats so you don't have to bring one (or two) from home. The grocery stores are amazing and you will find almost everything you get from home there.
Bring lots of very strong sunscreen as it is much hotter than people realize. Another suggestion, is to bring small life jacets for the kids. This is the safest way to bring the kids in the water (this is what I did when my kids were small, I packed them in our suitcases) and if you wanted to go on a boat tour you would have proper fitting life jackets as I am sure no tour company would carry the infant size life vests. You can still place the kids in a water ring but if they move around or slip you know they are the safest.
Good luck and enjoy!