Claire Brown, communications director of the Smithsonian Air and Space Museum, was asked the same question.
“The reason that it’s not in this museum was that it was personal
property of Alan Shepherd. In other words, he took it to space, he
brought it back, it was still his personal property he donated it and
it was his. That’s the reason,” said Claire Brown, the Smithsonian
National Air and Space Museum’s communications director.
“Things were a little different back then. You could take a certain
amount of personal property. There are different rules now, but at the
time that he did it, he was able to bring his own personal club.”
At first glance, it seems a bit odd that the club could be considered "personal property" since the shaft was actually a contingency sample extension handle. But as Shepard explained, "That [contingency sample extension handle] was already up there to be thrown away."
In 1972, Bing Crosby — who had played golf with Shepard, and happened to be a board member of the USGA Golf Museum — wrote Shepard asking if he would be willing to donate the club. He agreed, and he presented it to the museum in 1974.