Nascar chief executive Brian France, 52, is sitting in his lofty corporate offices overlooking midtown Manhattan. His gray suit and glass-paneled conference room give little hint that at the heart of his organization, cars careen around a racetrack at over 200 miles per hour in front of tens of thousands of fans.

This weekend, he’s gearing up for the first Sprint Cup series race of the season in Sonoma, Calif., where he will host employees from technology companies such as Facebook and Twitter. The outreach is part of an effort...