@rrcrawford
What you have is the sale of a personal asset which is treated as a capital asset. You report it on Schedule D and Form 8949 (the same place you report the sale of stock you sold in your brokerage accounts). Form 8949 reports the details of the sale (dates and amounts) and Schedule D summarizes them and combines with other related Schedule D transactions.
Since you owned it longer than a year, it is long term. The sales price less your cost is your gain or loss. A Gain is taxable and your loss is not (if you end up with a loss, and the auction house reported the sale, make the sales price match the amount you received and they reported to the IRS and your cost basis the same amount so it generates no loss on form 8949. This feeds the IRS computer that you reported the sale and avoids a notice and does not take any unallowed loss.
Your cost basis is the cost you purchased the vehicle for many years ago plus repairs and modifications that you made to the vehicle. The value of your labor is zero, but if you paid a mechanic to do repairs, they will be included in the cost. The commission of the auction house is added to the cost as is shipping to the auction or to the buyer.
Here is a link to the IRS webpage detailing the rules.
Tax Topics - Topic 409 Capital Gains and Losses
I would ask the Auction company if they will issue any IRS reporting forms. I would anticipate they will issue you a 1099 as that is their support of their deduction.
As to the receipts, it would be good to have them. I would make a spreadsheet and list each expense with the dollar amount and mark if I still had the receipt in a column. As long as you can show the majority of the receipts, the IRS will allow them in the event you are audited on the transaction. If you have no receipts and guessed, they will disallow.
I would not report it on Schedule C. That would be used if you were in the business of buying and selling cars and were not incorporated.
All of this being said, most people sell their used cars (and garage sale items) and never report it as they are at a loss and the amounts are minor.
My UserName should give you an idea as to my profession