From the Padre Island National Seashore website:
There are no public RV hook-ups anywhere in the park, but an RV dump station and a water filling station are available for all campers staying in the park.
Check out time for Malaquite and Bird Island Basin is 12:00 noon.
Camping is permitted only in the five camping areas listed below. Please click the link for the individual camping area for more details.
Bird Island Basin
I would call the ranger station for more specific advice.