Paris to Corbigny -- Train Strike?

Have you tried checking using Rome2Rio.com? Can you rent a car? This may be one of those times when driving yourself is the best, maybe even only, choice.

Is there a rental car company that will let us drop off the car in or near Corbigny and if not, how close can we get and are there taxi connections?
It sure would be more simple if the trains are running!

Yes, it would be a whole lot better if the trains were running. I do not know of any car rentals in Corbigny. I've only ever driven through on my way to someplace else. I just looked at the Flixbus route map and the closest spot that they serve is Auxerre. Maybe you can take the train to Auxerre and then take a local taxi to Corbigny -- you will probably have to pay for a round trip but if you have to leave the car in Corbigny that could be an answer. Or you could see if there is a take a local bus from Auxerre to Clamecy or another town near Corbigny. Information on the local busline line is at http://www.yonne.fr. Click on Territory and the click on Transport from the drop down menu. I don't know if this bus service is run by SCNCF and so is subject to the strike. The thing is though that with the trains strike some trains may be running so maybe you could get to Corbigny by train but you will not have real clarity on that until you get to the station and who wants to risk that. Have you contacted the Corbigny tourist office to see if they have any ideas?

We just returned from France last Thursday, and our experience in Provence was that no trains were running on the strike days. They did honor our tickets the day before a strike (twice), but it meant changing hotel reservations and having to pay for nights we didn't use in pre-paid hotels. They had said that some trains would be running on the strike day, even as late as the day before the strike, but we found that on the strike day our original trains we not running, so we were glad we had decided to go early on non-strike days. Also, we had rented a car for a couple of days that we turned in on a strike day at the TGV station on the outskirts of Avignon, thinking we could take the local shuttle train or bus to the city center station. It was a ghost town - no trains, no shuttle bus. Only way into town was by taxi. We had a wonderful trip, but the strike added an extra stress factor and expense we hadn't anticipated.

The major rental agents (Hertz, Avis, Europcar) have offices a few towns away, such as in Auxerre, Le Creusot, or Avallon. Searching on Google Maps can be the easiest way to start. These smaller agencies are shown in Corbigny itself and also have offices in Paris:

https://www.ada.fr/location-voiture-paris.html

https://www.rentacar.fr/agences

Thanks for all the advice. I think we will just rent a car and drop it off in Auxerre -- spend the night there and take a cab the next morning to Tannay. It will be a bit pricey but probably the most reliable way to get there.

We actually need to be in Tannay, not Corbigny. We were originally advised to get lodging there but I can cancel my reservation now. We will be picking up a self-hire boat in Tannay Sunday morning.

Oh well, that is something completely different. I wouldn't stay in Corbigny anyway if I needed to get to Tannay and was leaving from Paris. You will be going past where you need to go if you go to Corbigny. Your solution is a good one.