Not bidding as such but AFAIK Finland's TASO system is built on Torneopal's platform
https://www.torneopal.com/ - albeit probably heavily customised after years in action.
(I've worked on tournaments that have used Torneopal that have had the same referee "marketplace" we have in the Finnish footy set up...)
Part of the system is a "marketplace" for referees. We refer to it as "buying" and "selling" games but there's no bidding or buying as such.
The system is quite incredible. I can see my whole match history and financial history, all payslips, assessor notes, fitness test results etc etc... but the "marketplace" is probably the most "valuable".
Of course, I can see my upcoming matches in one view. But, if I can't make a match (or a sequence of matches on the same day grouped together) then I can just hit the "sell" button. I can choose to "sell" it to a specific colleague (they will have to accept next time they log in) or to have it put on the "marketplace" (which we call something like "openings" in Finnish.
The marketplace has a list of all the available games that other refs/ARs want to "sell". I can search the list by location, division, pitch etc. I can see who the teams, other refs, selling ref etc. are and I can click to buy. I get an error if I am not qualified.
Of course, we also have a F***book group where we beg each other to take difficult games, and some off the system friendlies get posted. Rarely, on really hard weekends when there are lots of spare matches, someone will offer extra cash if someone else will cover a game they have been scheduled.
AFAIK all out scheduling is done in the same system. We state how many match events we want per week, we can block days/times and then our masters work their magic. I don't know how much Torneopal's system helps scheduling with location data etc. (but it does automatically give us our milage based on GPS data of our homes and pitches)
Your chum should get on the blower and get a demo from Torneopal