WASHINGTON — A Twitter post recently caught the eye of Bill McKibben, the environmental advocate and godfather of the Keystone XL pipeline protests. It included an image from “The Simpsons” showing Homer and his family basking in mountains of cash in their living room, followed by a report on Hillary Rodham Clinton’s appearing at a fund-raiser with a lobbyist from the Keystone fight. Mr. McKibben’s environmental organization, 350.org, has been trying to raise awareness about the ties it sees between lobbyists for the oil pipeline and former aides to Mrs. Clinton. He promptly shared the post with his 150,000 Twitter...