Need help ID'ing this vintage mid-century upholstered chair  

I don't know who made it or designed it, but I've heard that general shape referred to as a Bergere (from a much earlier era, I know--I just mean the barrel type with that particular shape).

I reupholstered a pair of very similar ones once years ago and it was very, very hard to do well. Not a project for a beginner or even someone with experience doing more basic furniture. Getting the fabric smooth with even tension on the inside back, and smooth around the back posts where they meet the seat in the back is tough to do. The inside back perimeter at that top is also tricky. You have to staple the fabric to a very narrow strip of the frame, then trim it to within 1/4" or less of the staples and then tack or glue double welt over the staples so that it completely covers the raw edge of the fabric too---while keeping the welt neatly within the recessed groove. Might wanna consider having a pro do the upholstery part unless you've done this specific type of job before and are comfortable with it. (or at least buy a junk chair first to practice on.)

You don't need to find an upholsterer who does mid-century modern furniture. In fact, it would probably be best to get someone who does a lot of antique or tradtitional furniture since these techniques are way more common in that stuff. Just make sure he or she is clear on the thickness of padding needed---less than on traditional furniture. The photo you posted is a good guide.

Looks like Ward Bennett for Brickel

And recently at Herman Miller.

https://www.hermanmiller.com/products/seating/side-chairs/landmark-chair/

https://www.hermanmiller.com/stories/why-magazine/ward-bennett-at-100/