Tesla Inc.’s top engineer overseeing vehicle development is taking a leave of absence from the company at a crucial moment when the electric-car maker is struggling to boost production of the Model 3 sedan, according to people familiar with the matter.

Doug Field, Tesla’s senior vice president of engineering, is stepping away from the company for several weeks, these people said. One person described the absence as a “six-week sabbatical,” and Tesla declined to say when he would come back.

“Doug...