WASHINGTON—A consulting firm hired by Burisma Group mentioned that former U.S. Vice President Joe Biden’s son served on the Ukrainian gas company’s board so the firm could leverage a meeting with the State Department, according to documents and a former U.S. official.

The documents—email exchanges between State Department staff members made public this week—show that the consulting firm, Washington-based Blue Star Strategies, used Hunter Biden’s name in a request for a State Department meeting and then mentioned him again...