There is the “historic” pleasure pier (pictured above) which was built in 2012. It might have a souvenir shop, but it’s mostly rides and food service. Here’s what Wikipedia says:
The new pier complex is located where the original Pleasure Pier stood from 1943 until 1961, when it was destroyed by Hurricane Carla. The original Pleasure Pier featured rides, an arcade, an aquarium, concessions, a large ball room, named the Marine Ballroom, and fishing at the end of the pier. It was also the site of the USS Flagship Hotel, an over-the-water hotel built in 1965 that was demolished after Hurricane Ike in 2008.
There are also some shops along the seawall which extend out over the beach but not over the water. Here’s a shot from Google Street View:
Looking further down the seawall I found a gift shop that IS over the water. Street View: