To the OP....without a couple of pictures or a description of what is included, I wouldn't be able to hazard a guess.
I've been through this a couple of times recently, although not with tools so much.
First thing I would do would be to get the heirs to look at the items and see who might have a want or need for any of it. Try to get rid of as much of it as possible my divvying it up among the heirs. Then bring in other relatives who are not heirs and see what they might want.
If you're going to do an auction, that will likely be for all items not just tools. After going through that process a couple of years back, I was left with thinking it was sort of a waste of (our) time. The auction service fees ate up most of the revenue from the auction. I wouldn't go that route again. Going through that process delayed getting the house empty and on the market by a couple of months. The taxes, utilities and missed interest (based on the home's eventual selling price) for those 2 months was more than the heirs received from the proceeds of the auction.
FB Marketplace or a yardsale would be a better way....but you'd have to find someone who is willing to do all that work.
In both cases I've been involved with, tons of valuable items ended up donated to Goodwill and lots more ended up in dumpsters with distant heirs saying "I can't believe you gave this away and threw that away"....but those that are actually doing the work just flat out exhausted from it and a little put out by criticism from those that will get a share of the proceeds but aren't doing any of the work.