No. This suggestion is, I'm sorry, utterly ridiculous. Even if it were possible---which it isn't due to the fact that many patrons have their emails long before they come to the library---but it would amount to get another public subsidy of a private mega corporation in the way of free tech support and data storage. Not to mention the security concerns, though to be honest I think that's secondary in the face of losing access to housing and benefits.
Or is SMS the only possible second factor authentication method?