Yeah you guessed right, as you're running the site off IIS it will use that applications identifier when checking permissions to access a network share.
You could use something like this
https://github.com/mattjohnsonpint/SimpleImpersonation
Which allows you to impersonate a windows user to access the share and end up doing something like
var credentials = new UserCredentials(domain, username, password);
var result = Impersonation.RunAsUser(credentials, logonType, () =>
{
return System.IO.Directory.GetFiles( @"\\server\share" );
});