Saarinen Executive Chair with Walnut Legs Assistance  

Plug
Try McMaster-Carr, but you'll need to know what size you want.

Threaded Nut
You'll probably be looking for a threaded sleeve or a threaded T-nut of some sort. If you go to the hardware store with the machine screw and find a nut that fits perfectly, write down the thread pitch and diameter of the nut. That is the size of the female sleeve or T-nut that you will have to find.
I recall that on my chairs, those female sleeves are mostly hidden under the vinyl. So, I do not know exactly how they are attached or what they look like. My guess is that you'll have to carefully epoxy it in place while making sure that none of the epoxy clogs the threads. Don't

I took it to an uphostery shop
and they confirmed that it is a t-nut. I will have them do the work since they have to remove the fabric in order to create the new holes. I will explain further and I feel that this project will help a few of you out there as well. Thank you

Todd V
Can you post a picture of the bottom of your chair so I can see the pattern? Additionally, are all the legs around 14 1/2 inches or so or are the front legs taller than the rear? Thank you

Ok
Sure, give me a day or two.
I do think that the legs are different lengths because the fronts and backs are at different angles.

Great.
I really look forward to the photographs and measurements of both the front and rear legs. If this project is successful I believe that many will be able to utilize it when repairs or change is needed. Blessings

Ahhh, forgot to take...
Ahhh, forgot to take measurements but I did get a photo. You may be orienting the legs incorrectly. Each of the two leg units contains a front leg and a rear leg.
See the bolt pattern in the photo. (The front of the seat is at the bottom of the photo)
Obviously, these are metal legs. Are planning to make a new set of wooden legs? Sorry, I wasn't exactly sure of the original problem.