Hello William,
This should be possible, but would most likely require something custom. I can’t speak to what it would be, my technical-fu isn’t up to par on that.
This would depend on the shop software and how its authentication works or what the authentication is based on. OAuth? OpenID? Once you figured that out, then using the appropriate authentication settings/integration (either from the ones I linked to or you search for others), then this should be possible.
I hope this gives you some primer to get started with.