With SSMS on my local machine, is it possible to login to a remote SQL server on my VPN network via my Active Directory credentials?
As I said in my comment, I do not fully understand what your issues are.  

But I think the solution is to open SSMS as another user.  

Personally I have the SSMS icon in my task bar:.


Shit + R click
Select "Run as a different user"
Enter the account and password you would normally use on your office machine.


SSMS will connect to instances as if you where on your office machine and had opened SSMS. 

Note I assume you are working remote for possibly the first time.  There are a number of security barriers, that your organization may have in place to prevent you from connecting from home.  I can't address those, and this is probably not the site to address them.