Who invented the wallet?

I’m guessing his name was Bill Fold.
Here is an interesting wiki page.

The word “wallet” has been in use since the late 14th century…

The word pocket (n.) mid-14c., pokete, “bag, pouch, small sack,” from Anglo-French pokete (13c.)

There’s no telling who had a hand in it.

I’d say the form changed with the adoption of more paper money, as opposed to coins. For coins you probably want a pouch, but when there are bills of the same size, what we mean by a wallet works better (but no so much for coins).

My guess is it was likely invented by one or more women, who (particularly in the past) rarely got their accomplishments recorded and celebrated under their names.

The term “bill fold” originate in about 1895 from the term “bill folder.” I think @Zaku and @Pinquidchance have it pretty well described.