Solved it. Was seeing some messages similar to those described here:
Using the SQL queries described in that post, I found that some user ids had been deleted from the main table but were referenced in other tables (particularly in reviewing).
I created a new user and then copied the id of that user to the rows of the reviewing tables containing deleted user ids. This substituted the new user for the ones that had been deleted. Once I did that, the submissions page loaded properly.
I also made a note on the affected reviewing rounds of what had happened. The real reviewers’ identities were still visible in the email history (invitations to review), so mostly the data loss was just a nuisance once the pages were loading again.