If the auction has already ended, then yes there is a formal process for sending an offer to any underbidder who hasn't opted-out of receiving them:
https://www.ebay.com/help/selling/listings/selling-auctions/making-second-chance-offers?id=4142
If the auction hasn't ended yet, then you can just cancel the bid that was placed by the bidder who doesn't want it any more, so the auction can proceed without them:
http://offer.ebay.com/ws/eBayISAPI.dll?CancelBidShow
If you cancel an active auction after it has received bids, then (1) you will anger the bidders and they might not come back to bid on your relist; and (2) you will probably be charged a penalty fee which is equal to what the final value fee would have been if you had sold the item to the high bidder.
https://www.ebay.com/help/selling/listings/creating-managing-listings/ending-listing?id=4146
