Cape Canaveral, Fla.—SpaceX’s first private flight streaked into orbit Wednesday night with two contest winners, a health care worker, and their rich sponsor, the most ambitious leap yet in space tourism. It was the first time a spacecraft circled Earth with an all-amateur crew and no professional astronauts. “Punch it, SpaceX!” the flight’s billionaire leader, Jared Isaacman, urged moments before liftoff. The Dragon capsule’s two men and two women are looking to spend three days going round and round the planet from an unusually high orbit—100 miles (160 kilometers) higher than the International Space Station—before splashing down off the Florida coast this weekend. It’s SpaceX founder Elon Musk’s first entry in the competition for space tourism dollars. Isaacman, 38, made his fortune with a payment-processing company he started in his teens. He’s the third billionaire to launch this summer, following the brief space-skimming flights by Virgin Galactic’s Richard Branson and …