LANSING, Mich.—Lobbyist Jason Geer pecked away at his smartphone, trying to quickly raise information on a lawmaker he was meeting about energy policy. “This is what things have been like lately,” he said, jumping into his car. “It’s just all phone calls, messages and meetings. One after another.”

Days are busy not only for Mr. Geer, who represents the Michigan Chamber of Commerce, but for a new army of lobbyists and interest groups swarming state capitals around the U.S. Businesses and industries seeking relief from the gridlock...