Weber was born in Pearland, Texas. He attended Alvin Community College in Alvin, Texas, and earned a bachelor's degree from the University of Houston-Clear Lake. Weber owned an air-conditioning company from 1981 until 2016. He was a member of the Pearland City Council from 1990 until 1996 and a Texas state Representative from 2008 until 2012. Weber was a grand jury commissioner for Brazoria County, president of the Brazoria County Cities Association, and chair of the Pearland Area Republican Party Headquarters.