It seems notable that an earlier cited use of an "X" to denote gender neutrality is in the honorific Mx., which dates in writing to the 1970s.
According to this article in The Huffington Post, Latinx appears to have grown into use in the 2000's, and it appears from articles covering folx that it was also cited in writing as recently as the 21st century.
The OED has this to say about the etymology of "Mx."
Apparently < M- (in Mr n., Mrs n.1, Ms n.2, etc.) + X n., probably denoting an unknown or variable quantity (compare sense 3 at that entry)
The referenced sense 3 refers to "X" as it is commonly used in algebra to refer to an unknown entity, and allusive extensions.
Katherine Rosman in The New York Times describes the earliest found citation of the honorific:
The first citation of Mx. found by Ms. Martin’s team dates to 1977, in a publication called The Single Parent. In the midst of the Ms. era, an article in it wondered whether a courtesy title that masks gender might help ameliorate any bias against single parents. “On second thought, maybe both sexes should be called Mx.,” the article said. “That would solve the gender problem entirely.”
This leads me to believe that "X" as a gender-neutral particle originated with "Mx.," functioning as a wildcard character of sorts, and was used similarly by the communities that coined "Latinx" and "folx."
It's possible that "folx" evolved independently of these other words that use "x" to denote gender-neutrality, but it would also be a significant coincidence. For this reason, I suspect that tracing earlier uses of gender-neutral "x" is the best we can do regarding an etymology of "folx."