Is the second "people" appositive?
Yes, I would consider it an appositive. It is possible that the comma replaces the conjunction "and", but I think that that is unlikely in this case. (A series of only two items usually uses "and" to connect the items instead of a comma.)
Could it be changed to "We’ve had a couple of incidents of people who thought they’d never need a food bank coming in very distressed"?
Yes, they have the same meaning. It seems to me that the people coming in very distressed are exactly the same people who thought they’d never need a food bank (assuming that the comma doesn't mean "and", of course). (I disagree on this point with some people who have written otherwise on this page.)
Who are "very distressed", we or the people?
The adjectival phrase "very distressed" is far from "we", with several other words between them. Furthermore, it seems to make much more sense that the people coming in would be distressed rather than the speaker's group (presumably a group that operates a food bank). I therefore think that it's safe to say that it modifies "people".