Elmo Shropshire spends most of the year setting running records. During the holidays, he gives his legs a break and but keeps working his lungs by performing his hit Christmas song, “Grandma Got Run Over by a Reindeer.”

At 83, Mr. Shropshire moves as if he could almost outrun a reindeer. His time of 29 minutes and 44 seconds earned him first place in his age group at the 2019 Masters 5K Cross Country National Championship. And after clocking a time of 7:42 at the One Mile Bang race in Los Gatos, Calif., in June, he now ranks...