Speculation, but I’m assuming there’s a direct line from haxxor to doxx(er) to vaxxer.
Leetspeak, the affective use of creating spellings in hacker communities, has taken up the use of -xxor or -xx0r as a creative spelling of –cker; thus, haxxor for hacker. haxxor – Wiktionary. The duplication of <x> is an affectation. In fact the very use of <x> instead of /k/ is an affectation: it is misusing the plural hacks (hax) as the stem of the noun hacker (†hacks-er) instead of hack. The spelling haxx – Wiktionary is also used for hacks and hacking, and I’m assuming it is a back formation from haxxor.
dox – Wiktionary is the Leetspeak respelling of docs (abbreviation of documents) in the sense of “Documents, especially information sought by hackers about an individual (address, credit card numbers, etc.)”
When it came time to form a verb out of the action of seeking dox on someone, the spelling doxx – Wiktionary prevailed. I am assuming it was done by analogy with not just haxx (still a noun), but also the verb *to haxx implicit in haxxor. Hence why the noun dox is more often spelled with a single x, and the verb doxx with a double xx (although both spellings are used). And I think this was affective analogy; I don’t think you needed the intermediate step of doxx0r (document hacker) for that to happen.
Now, English has a longstanding spelling convention that when you add a suffix to a word ending in short vowel + consonant, you double the consonant to keep the vowel short. Sit > Sitter. Spam > Spammer. You could say that by analogy, dox > doxxer. But there was no use of xx like that before in English: box > boxer.
Nor am I convinced that the need to make the vowel before the <x> short played a role: I just don’t think leetspeak would have ever spelled baker as baxor, for example, so that you would need to spell backer as baxxor to contrast the vowels.
So what I think happened is that the –xx spelling of haxxor and doxx opened up the possibility of applying the consonant doubling rule to other verbs. Once doxx was in place as a verb, and haxxor was already there, it became easier for verbs ending in /ks/ to be spelled with an xx, and even more easy for –xer nouns to be spelled as –xxer—as long as they are analogous to doxx: that is, primarily used online and informally, and clearly abbreviations of of something else. In fact, the parallel of haxxor made it possible to go straight from dox to doxxer, without even positing an intermediate verb to doxx—you can just draw an analogy now with spam > spammer.
The adjective is anti-vax – Wiktionary. By analogy with doxx and haxxor, and for that matter spam > spammer (a rule now applied to anti-vax by analogy with haxxor): anti-vax > anti-vaxxer – Wiktionary.