In the following sentence, should I include a comma after "acquiring"—and if so, why?
German residents who agree to declare that they possess, or have begun the process of acquiring**,** documents...
Yes, you should. The phrase "or have begun the process of acquiring" is a parenthetical expression. The portion between the two commas can then be omitted without changing the meaning of the remaining sentence.
Have a look at this (found via googling), rule 5.
I don't interpret the sentence the way Lawrence does (in a separate answer to this question). Rather than viewing the sentence as beginning with this (relatively simple) baseline clause:
German residents who agree to declare that they possess documents...
attended by this parenthetical aside:
or have begun the process of acquiring
I see the sentence as beginning with two categories of equally significant and valid subjects, most likely in this pairing:
German residents who agree to declare that they possess documents
and
German residents who have begun the process of acquiring documents
Subordinating one class to the other makes no sense as a matter of categorization. I agree with deadrat's comment that the way to express the equality of the two subjects is to revamp the wording, not simply to add or delete a strategically placed comma.
My preference would be for a wording like this:
German residents who agree to declare that they possess documents or who have begun the process of acquiring documents...
(if that's the meaning you intend to convey) or
German residents who agree to declare that they possess documents or that they have begun the process of acquiring documents...
(if that's the meaning you intend to convey). There are many other possible ways to clarify the meaning and the relationship of the dual subjects of the fragment, of course; and the best way of handling the sentence as a whole depends to a considerable extent on the content of the unseen remainder of the original sentence, among other variables.