After doing some more research, I uncovered some information that might serve as a partial answer. I hope somebody else knows more, but I'm posting this so that other people don't have to go through the same stuff I did.
The terminology does in fact seem to be derived from "moderator." I found this page describing the moderation system of SoylentNews that uses the terms “modder” = “moderator,” “modding” = “moderating” = “voting on posts." It doesn't use "upmod" or "downmod," but “modded down” and “modded up” are used to refer to the comments that had their score decreased or increased by the actions of a moderator.
SoylentNews uses Slashcode, and I found references to "modding" posts in this page of SlashDot .sigs from 2003:
"I'll likely get modded down for this but..." - Slashdot Karma Whores
“Usually at karma-cap, please upmod my troller account instead.”
So far, the term seems to be connected most strongly to the internet community Slashdot. I also found an article that suggests this, "Slashdot and CmdrTaco — the end of another geek era" by
Mathew Ingram, which says:
Slashdot’s “upmod,” a vote for a smart comment or thread, was the
dominant geek status symbol, a predecessor to the “digg” vote and the
Wikipedia says Slashdot was founded in 1997, but the site's info page on "Slashdot Moderation" says that the moderation system was introduced some time after the site was established. Unfortunately, it doesn't give an exact date.