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? I'm trying to login to a SQL server on my work network with a VPN connection to the network and by using my AD credentials. 
When I'm directly on my office machine, I use "Windows Authentication" as the Authentication method in SSMS and that works fine.

When I'm VPN'ed onto the network and using my local machine, Windows Authentication doesn't work (understandably) but if I try to use the "Active Directory - Password" Authentication method (I was hoping I could put in my work's AD account to connect to the remote server) I get the following error:



It sounds like it can't find the server, but when I ping the server from my local machine, I'm getting a response.

Note: "Allow remote connections to this server" is checked.
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.