Democrats have seized on a campaign strategy that they believe combines the best of patriotism with progressive bona fides: enlisting military veterans to run for Congress in the midterms Nov. 6.

Jon Soltz, chairman of VoteVets, a political-action committee dedicated to pushing military Democratic candidates, is surveying dozens of midterm contests to help candidates with funding to match GOP-aligned campaign spending. Republicans counter that time spent in uniform can’t mask that candidates are still Democrats and will face...