Almost no beachfront rentals on Siesta Key allow renters to bring their small (or any size) pets with them. Also pets are prohibited from all Siesta Key beaches by local ordinance.
The reason for this is nearly all of the beachfront rental units here on Siesta Key are individually owned units of large condominium resorts and typically the by-laws of those condominium resorts either don't allow pets at all, or restrict them to owners only (no renters). These "Condo Rules" are strictly enforced.
Here is only one direct Beach Front Condo (that in past years) allowed pets, if this is important, contact them directly to see if they still allow pets:
For most visitors with pets, your best bet is to find one of the many privately owned rental homes (off of the beach) where the owners permit pets. Often these rental homes may only be a short walk to the beach. The highest concentration of rentals allowing small pets and a short distance from the beach is in the Siesta Village area, so if you're determined to bring your doggie, start searching for home rentals near Siesta Village.
There are some Bay Front private homes that allow pets, here is an example of a nice rental that allows pets: