Running corporate in-house legal departments like a business is quickly gaining traction, says Connie Brenton, chief of staff and senior director of legal operations at NetApp Inc.

One example of that trend, she said, is the membership growth of the Corporate Legal Operations Consortium (CLOC), a group dedicated to helping legal operations professionals. CLOC, which became a nonprofit in 2016, has nearly 1,300 members worldwide.

The increasing importance of the profession reflects evolution in the legal industry, including changes in...