In case others run into this, I think I was able to resolve it. This happened after clearing out my preview branches, but I think hitting my branch maximum triggered the issue. It seems like I was able to clear the issue by going in to the vercel project I was having the issue with and saving the Neon connection again. It took a minute, but I found the connection setting in Storage, clicking on my Neon instance to open the "Getting Started" page for that Neon connection (even though it's been connected for many months), then navigating to "projects" in the left rail. Here, I was able to select "Update Project Connection" from the snowman menu in the "Projects" table on the page, change nothing in the dialog, and click Save Changes. This seems to have reset the connection and things now build again.
So, if you're following along: Your Vercel Project > Storage > [your Neon instance] > Projects > Projects > Update Project Connector > Save Changes.