There are no facilities on the park it self to spend the night, but camping on the beach is a growing trend.
camping
Camping on the beach is not allowed. It was until 2010.
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.