My Macbook Air 2014 is stuck in an infinite boot loop because the SSD is not registering

If the SSD is not showing up in Disk Utility then it is likely that the SSD has failed or the hardware that supports the SSD has failed. The failure is likely coincidental to the OS reinstall and sale, but it can't be ruled out that the purchaser did something to the Mac. What that might be I couldn't tell you.

Realistically the only way to diagnose this is to open it up, preferably by a certified Mac tech, and see what is going on.

You are probably looking at having to replace the SSD (I think this Mac has a removable SSD) in your Mac.

You should normally be able to boot up the OS from an external USB drive.

The SSD is about the only easily replaceable part inside: (once you've got the correct screwdriver shape for the case screws).

It's worth checking that a) it's actually there, and b) seated correctly. After that, you can check whether it has failed only by replacing it.

There are original new or secondhand Apple OEM modules on eBay; or you can get third-party units made by OWC and others.

This webpage tells you everything you need to know about which of the many Apple SSD modules will fit. https://beetstech.com/blog/apple-proprietary-ssd-ultimate-guide-to-specs-and-upgrades