Get and got both work here but require the sentence to be parsed differently.
What your copy-writer apparently intended was for the sentence to be parsed like this:
Our small initiative turned into a sizeable movement with the support of influencers, who helped
(a) spread the word
(b) get people to make a pledge by greenifying their Facebook profile pictures.
With this parsing, get is the correct form, namely a bare infinitive parallel to spread. In your first approach to the sentence, however, you succumbed to the temptation to parse it thus:
Our small initiative turned into a sizeable movement with the support of influencers, who
(a) helped spread the word
(b) got people to make a pledge by greenifying their Facebook profile pictures.
With this second parsing, got would indeed be the correct form, namely a simple-past indicative parallel to helped. The copy-writer apparently did not intend this second parsing and failed to appreciate how tempting it is. This failure is thoroughly normal: when we have just recently written a sentence we know what we meant and parse accordingly, but a careful writer makes extra effort to assure that a reader will, on the first try, parse as intended (or as needed, which may not be exactly the same thing—more on that anon).
For this purpose, one good approach is to reread one’s sentence after the passage of some days, and so discover the trap by tripping on it, after which one can correct or defuse it. If time does not permit that, one may show the copy to someone else (such as a wonderfully intelligent voice-over person), pay attention to that person’s mis-parsings, and then reword so as to prevent them. Best of all, one should learn to reread one’s copy, however soon after writing it, as if one had never encountered it before.
The syntactic booby-trap in the present sentence could be defused by simply changing the bare infinitives to to infinitives (which are a little wordier but perfectly acceptable here) in order to make this less of a garden-path sentence:
Our small initiative turned into a sizeable movement with the support of influencers, who helped to spread the word and to get people to make a pledge by greenifying their Facebook profile pictures.
One disadvantage here, though, is the stacking of to-infinitives: to get people to make. That’s a bit ugly. But frankly I fail to see why your copy-writer’s intended parsing is in any way preferable to yours. To get people to do something and to help get people to do something are finally not very different in meaning. So why not just go with the second parsing and use got? Sometimes one’s reader, following the garden path, winds up with a better sentence than one originally intended, and the wise writer will humbly embrace that one.