There are no facilities on the park it self to spend the night, but camping on the beach is a growing trend.
lodging
The nearest hotels are in the town of Pedernales. They're not luxury hotels, but have enough to be comfortable. Most hotels in Pedernales have A/C, hot water, Cable TV, pool, 3 meals, etc., for less than $40 USD per night per room.
camping
Camping on the beach is not allowed. It was until 2010.