A lapse in federal funding is snarling the Equal Employment Opportunity Commission’s workplace discrimination litigation, and the side effects of court delays and scrambling personnel could linger long after the government reopens.

The EEOC can have several hundred bias lawsuits pending in federal district and appeals courts at any given time, based on agency litigation statistics. In the months leading up to the partial shutdown that began Dec. 22, it sued companies such as Walmart Stores Inc., Charter Communications,...