One assumes there'd be a transition phase while all this was designed and implemented and sold or at a minimum a certain number would have to be sold to reach quorum. I know assumptions aren't great ideas in what is already a pretty improbable case but still...

One is working on a timeline since things need to keep going and 300 million seats being auctioned sequentially would require 10 per second be sold in under a year, even population birth-death values would keep that number rising at faster than one a minute after that. So presumably you'd either number them and auction them all off simultaneously on vot-eBay or in lots at a rate that allowed the boards of elections to verify the buyer was an actual qualified person owning no others. A one year transition where roughly a million went up every day would probably do the trick.

If people wanted live auctions, and it were limit one per customer, you could divide all or most of them into each of the roughly 3000 US counties, then let each board of elections handle their average 100,000 each. They could actually auction the things off classically in under a year without having to do it 365/24/7. I could see turning it into a bit of a fundraiser that way, at least for the cool numbers, or at least at the bake sale level, but if it is one per customer as long as the number of votes paralleled or exceeded the number of people who might be willing to buy one at all you're never going to have a real market for them except the 'hey, first eight digits of pi!' or '04-7-1776! $10,000 for independence day!' novelty ones.

I could actually see this working more effectively and more morally if you handed every 18 year old (or whatever) a serial-numbered token coin good for 60 years at a 78 year life expectancy card which they could then re-sell or not as they saw fit, coin goes into machine, vote screen comes up. That might actually make an amusing short story, slightly cyberpunk mildly dystopian setting, exploring how the vote tokens might turn into the new currency with mili-vote through mega-vote denominations and the means of federal taxation via issuance. hmmm...