Welcome to this forum.
Re: “I’ve been coding Fortran professionally for about a year and a half now, but the legacy code doesn’t use any features of modern Fortran, so I’m very new in this regard”, hopefully you will find the Learning section at the parent site of this forum, fortran-lang.org helpful:
Learn — Fortran Programming Language
Re: “I’ve struggled to find documentation on the “module procedure” statement, and interfaces in general” please note with Fortran, as things stand now, a good place to start for Fortran language semantics is the books (also as e-books), especially “Modern Fortran Explained”. Also for developing modern applications, consider the book by @milancurcic
Immediately for “module procedure” you can consider compiler vendor sites and their developer reference guides such as with Intel Fortran and IBM XL Fortran .