Yes, UNC paths are supported. However accessing network shares (UNC paths) is different than local folders in regards to user context.
By default, ASP.NET applications run under the context of NETWORK SERVICE or ASPNET users but when accessing a network resource, the other machine will not recognize those users as they are local users. So you need to run the application under the context of a common user which you can add to the ACL of the network share with the required permissions.
If there is a domain, you can use a domain user for this purpose. When you include the below settings in web.config:
<authentication mode="Windows" />
<identity impersonate="true" userName="SomeUserX" password="Password123" />
The ASP.NET application (and so FileVistaControl) will be executed directly under the identity of SomeUserX and not under system accounts like NETWORK SERVICE or ASPNET. So, you need to make sure SomeUserX can access the share. If you are connected to a domain, this would be easy because you would go to the other machine and give permissions directly to this domain user as you can select him in the list.
However, you can not select and add a user of another computer if there is no domain. In this case you need to create a user with same name (eg. SomeUserX) and password on the other server and use this common user for the impersonation setting above. FileVista is dependant on Windows permissions so you should first try in Windows if you can access the share with the user SomeUserX that is created on both machines. If Windows can access then FileVista will be able to access too.