I am trying to find an alternative word for acquisition in the context of acquiring a license to a piece of software. I would prefer a word that's shorter to put on a report than the "Number of License Acquisitions."
It sounds as though you're trying to come up with a brief wording for an entry in a report that identifies the number of licenses that consumers have acquired from a software company.
If so—and if the primary audience for the report is the software company—it makes sense to frame the wording not as "Number of License Acquisitions" (which looks at the number from the point of view of the licensees), but "Number of Licenses Issued" (which looks at the same number from the point of view of the company). And if you're trying to be concise, there's no need to say "Number of Licenses Issued," because "Licenses Issued" says the same thing.