Exxon Mobil Corp., once a powerful skeptic of global warming, will now be among the first oil companies to put money into the fight to make climate change a political priority in Washington.

The U.S.’s largest energy producer will commit $1 million over two years to promote a national tax on carbon as a way to address the environmental issue. The funding will back an initiative designed to appeal to the Republicans who now control Washington, and may open the door for Exxon’s peers in the oil industry to follow.

As...