Is there a regulatory requirement that a loan application must have the applicant's signature?

App was taken by phone and already closed.

no there is no requirement to have the applicant's signature. Applications are taken by phone or via internet all the time and don't have a signature. Simply note on the application that it was taken by phone and who took it.

In addition to what WS mentioned -

In many cases the secondary market requires signatures on applications, as a certification of facts. In addition, on the FNMA 1003, there are a dozen reps and warranties that the applicant vouches for when they sign the app. It is a very powerful tool in the courts if there was fraud.

I agree that it's not required by regulation but that they're often required by the secondary market. In those cases you can get the signatures on the application at closing if the application was taken by phone or online. You don't necessarily need to get it at the time of application.