"purchase access" is a correct and usable phrase in written English.
You can use it to refer to the act of obtaining access to something through purchase. For example: "We need to purchase access to the online database before we can start searching."
Non-subscribers can purchase access.
(Subscription required; non-subscribers can purchase access to the issue).
(The entire article is available to subscribers; others can purchase access to the issue online).
And universities would be able to purchase access to the entire collection.
If your subscription has expired, you may renew your subscription or purchase access to a single issue.
(I profiled Hansen in this week's issue; the full article is available to subscribers via the digital edition; others can purchase access or read an abstract).
They wondered where the door to which they had purchased access might take them, someplace in the mountains or on the plains or by the seaside.
The Band piece is full of crass vignettes of corporations and wealthy foreigners pretty straightforwardly purchasing access to Clinton with donations to the Foundation.
"The British Library purchases access to the Times online library" – and so out of copyright Times archive is kept behind a paywall, because it is difficult to copy, or even to access, piles of old newspapers.
Harvard University, which holds a competition among dormitories to register the most voters, is one of a growing number of schools expanding efforts by purchasing access to the Web site of TurboVote, a nonprofit effort that provides complete online registration and automated vote-by-mail services.
Magnatune's comprehensive pre-purchase access facilitates music discovery and allows an informed buying decision setting it apart from conventional online music stores.
