CAPE CANAVERAL, Fla.—Virgin Galactic finally has the federal government’s approval to start launching customers into space from New Mexico. Richard Branson’s rocketship company announced the Federal Aviation Administration’s updated license on Friday. It’s the final hurdle in Virgin Galactic’s yearslong effort to send paying passengers on short space hops. The company is working toward three more space test flights this summer and early fall, before opening the rocketship’s doors to paying customers. The original plans called for company engineers to launch next to evaluate equipment, followed by a flight with Branson and then a science mission by Italian Air Force officers. In the meantime, Blue Origin’s Jeff Bezos earlier this month announced plans to ride his own rocket into space July 20 from Texas. Virgin Galactic officials acknowledged the growing chatter over whether Branson will try to beat Bezos into space. “Clearly, Sir Richard Branson’s flight date has been subject …