If the government gave both companies the "GO" tomorrow (Monday, 14 May 2018), SpaceX would be first because they would probably have a launch pulled off by Monday, 21 May 2018. SpaceX has the rocket (Falcon 9) and the Dragon capsule. Blue Origin has a suborbital rocket (New Shepard) and a capsule, but they are still a little behind on the infrastructure to just launch.
Now, practically speaking, the government is not give a "GO" tomorrow. There is still a lot of regulation. And, there is a third player: Virgin Galactic, a company that has already launched people, albeit not tourists, to the edge of space on suborbital flights.
Space tourism is coming, and my prediction is this: by the end of 2018, someone will have paid a private company in the United States for a ride into space, likely suborbital. Whether that ride is with SpaceX (which would probably be orbital), Blue Origin, or Virgin Galactic, I cannot say. From a purely technological "rocket science" standpoint, Blue Origin could pull off a suborbital flight first because they could probably get government approval for that before SpaceX could get approval for an orbital flight, based on maturity of the respective technology. Virgin Galactic practically has approval, but they have had some setbacks recently.