Do adapters exist to install generic SSDs in MacBook Airs? I have a rescued mid-2013 11 inch MacBook Air working great off external USB 3 hard drive but missing its internal SSD.

They use a custom SSD, which seem to be expensive even used and aftermarket.

Are any adaptors available to install a generic SSD?

I Googled and thought I found that they do now exist, but then I realized the ones I found were the opposite, for using the SSDs from various MacBooks in other laptops etc.
The air has no extra room for internal parts. 

Unless the “generic SSD” is substantially smaller than the apple part, there would be no room for it. The adapter of choice would be an external adapter if you’re looking to cut costs. 

External connection also simplifies things like whether the storage interface and firmware and controller are compatible.