It happened around 4:45 p.m. Wednesday at the restaurant located at 3315 Palmer Highway.
The 65-year-old woman pulled up, so her children could get out of the vehicle and go inside the restaurant.
Shortly after they got out, she briefly lost consciousness and the SUV rolled forward hitting both children, Texas City PD spokesperson Allen Bjerke said.
The girl was pinned underneath the vehicle but was pulled out with the help of bystanders. She was taken to UTMB and may have broken her leg.
The woman's 5-year-old grandson was also hit. He was not injured. The woman was also transported to UTMB for evaluation.
The city health department and the Fire Marshal's Office plan to send someone out to check on the condition of the building.