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The face object is small (it’s the parameters). You can save that.

I did the first part of this (not the API part) a while back to generate avatar faces for minecraft with a random sequence of numbers (one per pixel, gray scale, and the faces were symmetric). I got some fun results as a wide range of anatomical facial morphologies were produced.

You could serialize the JS representation to Base64 or Base10 if you want it to be numeric.

But that's a lot more data than just storing which of the ~2^50 possible faces was generated. You could serialize the entire face including the colors and scaling factors into a much smaller string, or just take one number that is used as a seed for a random number generator that sets the other parameters

The seed version with random parameters doesn't allow you to or your users to design their own. Just random selection until you get one you like, no editing of details.