The ACME client only has to run once every 90 days, to validate domain ownership and retrieve an updated certificate. After that, the certificate can be stored on and loaded from disk - the same way most of the files being served are probably stored.
It's true there might be some added complexity, as the private keys will need to be stored securely. And session resumption data, if you need to support that. Doesn't seem like an insurmountable problem, though.