DONNA, Tex. — In this Rio Grande Valley town of trailer parks and weedy lots eight miles from the Mexico border, people call them runners or politiqueras — the campaign workers who use their network of relatives and friends to deliver votes for their candidates. They travel around town with binders stuffed with the names and addresses of registered voters, driving residents to and from the polls and urging those they bump into at the grocery store to support their candidates. Despite rumors that some politiqueras went over the line in encouraging voters, the tradition continued in Donna and other...