Attached is a diagram I have created based on the info you have provided so far:
This should help us make sure we are talking about the same printer enviornment moving forward. As you can see from the diagram your desired printer is sitting back in the corporate office. The only way to get to that printer is to be a member of that printers network, like you are when you are physically present at that location. The VPN provides your computer a direct link to the VPN server and makes the computer think it is physically on the corporate network. This should also be granting you the permissions you need to see and talk to the printer, but it only works when you are VPN connected. A normal internet connection from home would not allow you see the printer sitting behind your companies firewall.
If you want to print to this corporate printer from home while VPN connected then look up the IP Address of the printer. You can then work through your OS install a printer wizard to setup and create a network printer installation. If your printer is running off of a print server (not the EWS) then your installation will look something like //servername/printername. If your printer does not have a print server then you will have to do a direct IP installation with the assistance of the network printer installation wizard from HP. Use the normal full software solution for your OS and run the network installation. Type in the IP Address when prompted and the HP install wizard will do the rest.