Is there any way to replicate the user interface for the Console in a web application using the API? It isn't clear to me that this is possible.

There are 2 possible options, depending on your needs:
1 - You can develop your own administration form and call VG API to manage user accounts, grant them roles, etc...
You have an example (that you can re-use if you want) in VG Websample application:
log in to the Websample application with an administrator role and checkout the menu option "administration >> users".

2 - WebConsole is ready-to-use console manages user accounts, applications, role, permission sets.
Like the WinConsole, it supports multiple applications, as well as shared roles, etc...
Unlike the WinConsole, you can customize the presentation of the WebConsole
(use your own CSS, replace VG logo with your company logo for instance).