Solution: Ensure that the account you use to create your form/spreadsheet is linked to the organisation node that contains your GCP project and Cloud Run service. In other words, your account needs to be a user under the relevant domain.
In my situation, this meant that my account had to have the same domain as the organisation node that contained my project. But this might not be the case in your situation. I would recommend talking to your Google Workspace administrator or tech lead. If all else fails, you’ll very likely have to redeploy your project.
In my case it wasn’t sufficient that I switched accounts when re-creating my form/spreadsheet as Google kept switching me back to my main account when moving across different sections in Apps Script. I fixed the issue by logging into the correct account using an incognito window.