You can have a veteran designation printed on your Oregon driver license, instruction permit or ID card. There is no extra fee for the designation itself, though the usual original application, renewal or replacement fees still apply.
Ask for the designation at any Oregon DMV office when your license, permit or card is issued, and bring your discharge paperwork.
A copy of your DD214 or DD215 showing an honorable discharge, plus whatever the DMV normally requires for a driving privilege or ID card.