Email settings are usually somewhere in the database. You’ll likely find it in a table that gives you a good clue about where the setting can be changed. It may be wise not to just search-and-replace it directly since data can be serialized, meaning that if you replace the address with a different one that has a different number of characters, that breaks the setting. However, there are plugins that can do safe search-and-replace because they unserialize and re-serialize data.
If you don’t find the email address anywhere in the database, the other option would be in the files. Yes, you can download the site’s files, typically using an FTP client, sometimes using a host’s file manager tool. Then you would want to run a “find in files” type command to search for the email address, and again, carefully replace it. If you do find the email address in files, it would be better to take the time to convert that to a setting that can be changed in wp-admin, rather than having to edit any of the theme or plugin files directly.