LAREDO, Texas—Jessica Cisneros stood on the sidelines of a parade here and shouted at Rep. Henry Cuellar, the longtime Texas lawmaker she is trying to oust in the Democratic primary, as he rolled by in a black Dodge pickup truck.

“The people of the district deserve a debate,” the 26-year-old said, pointing to Mr. Cuellar, who was cheered along the Washington’s Birthday celebration parade route. Ms. Cisneros couldn’t formally participate—because while incumbent lawmakers are welcome, political challengers aren’t. She walked...