SPOKANE, Wash.—House Speaker Paul Ryan might not be the only member of the House GOP leadership not returning to Washington next year.

Rep. Cathy McMorris Rodgers, the No. 4 Republican in the House and the highest-ranking GOP woman, is locked in an unexpectedly competitive race this year in her eastern Washington-state district. Republicans are running on the recently passed tax cuts and the growing economy but face energized Democratic voters, putting seats held by longtime, powerful lawmakers into play.

Mrs....