How can I legally upgrade my MacBook Air from macOS X 10.7.5 to macOS High Sierra?

I would guess that Lion is too old to be able to access the Support article (https security changes) or the App Store any more.

Before you start, make a bootable clone backup of your Lion system, if you do not already have an up to date one.

If export controls are not an issue, try putting a copy of the .DMG containing the Sierra installer onto a USB drive. Copy it from there to the old Mac. Open it on the old Mac and upgrade the old Mac to Sierra.

Then the old Mac may be able to access the App Store link for High Sierra to go the rest of the way.

First question. Why can't you ‘access’ the App Store? That’s where you download full installers of macOS. Don't even think about downloading it from anywhere other than Apple. You will regret it.

How to download and install macOS - Apple Support

That MacBook Air is currently 14 major versions behind the current version of macOS (Tahoe).

Even if you got it upgraded to High Sierra, it would still be 8 major versions behind Tahoe. Vendors like Microsoft and Adobe support only "the most recent three", and even many Open Source developers have moved on – past High Sierra. (There are old versions of Firefox and LibreOffice that run on High Sierra … but the current versions require a minimum of Catalina, and LibreOffice will soon require a minimum of Big Sur.)

Then there is this:

The difference is even greater on multi-core CPU benchmarks, where the M4 MacBook Airs score at least 15 times as high as the 2011 ones.

That 2011 MacBook Air is now too old to support any newer release of macOS that will comply with current security protocols used by most web sites, and email service providers. Time to consider a new or Apple refurbished Mac whose operating system would not support any applications you are presently running on Lion.

I might guess export controls or trade sanctions or denied persons are involved here, and we can’t help with any of that.