This step only required filling out an online form, which took four days for the court to process. The court then sent me a copy of my court order via USPS, which took an additional three days.
In order to file for a name change in Multnomah County, all you need to do is visit the "Name and/or Gender Identity Change" webpage and—assuming you are an adult—click on the option that says [PACKET] - Name and Sex Change, Statewide, iForm, the first option in the "Local and Statewide Name and/or Sex Change Forms for Adults" table:
All you have to do is fill out the form and wait a few days for the courts to process it. It is refreshingly simple!
(The rest of the process will not be. 😭)