Hackers by Rick Cook, published in the April 1989 issue of Analog Science Fiction/Science Fact.
Part of a review from actsofminortreason.blogspot.com:
In "Hackers," we find ourselves under the sunny skies of Canaveral Field, where final preparations for 1989's Space Week are underway. This is where the rocket jockeys come together, from the San Antonio Rocket Club to the Los Angeles Space-Faring Society, with competitions as varied as a race around the moon and a mock rescue flight to one of the dozens of space stations orbiting Earth. In 1989.
Yeah, that's right. Because the 1989 of the nameless protagonist isn't the 1989 you and I lived through - it's a parallel universe, which split in the late 1950s when the Soviet Union failed to place Sputnik 1 into orbit. With the science of rocketry focused solely on ICBM development, and with no prevailing mood in the United States that the USSR was overtaking them, there was no space race. Governments didn't lay down billions toward getting men into orbit and, later, to the moon.
It was the amateurs that did that instead, as part of an entirely private space program - the moon was first visited not by the United States in 1969, but by the National Geographic Society in 1975... which actually makes a lot of sense when you think about it. By 1989, spaceflight is routine, and everyone accepts the world for what it is - except Crazy Eddie, a man who always hangs out around the rocket meets and never seems too right in the head. When the protagonist and his copilot launch in their Delta Doll on a mock rescue to a space station run by the Boy Scouts, they discover that Crazy Eddie has stowed away on their flight, and that his presence may make the difference between success and disaster.