Weber was born in Pearland, Texas. He attended Alvin Community College in Alvin, Texas, and earned a B.S. 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. He entered Congress on Jan. 3, 2013.