Where do chrome/chromium store search keywords?

The Chrome search keywords are stored in the Web Data SQLite file inside the User Data folder. The exact path depends on the OS you are using:

macOS

~/Library/Application Support/Google/Chrome/Default/Web Data

Windows XP

C:\Documents and Settings\<username>\Local Settings\Application Data\Google\Chrome\User Data\Default\Web Data

Windows Vista / 7 / 10

C:\Users\<username>\AppData\Local\Google\Chrome\User Data\Default\Web Data

Linux

~/.config/google-chrome/Default/Web Data

Inside that Web Data SQLite database file, the table keywords has the list of search keywords.

You can edit it using anything that can use SQLite databases (e.g. by hand in Microsoft Access with an SQLite ODBC plugin, scripting a solution with Python's sqlite3 module, etc.)

Unfortunately, Chrome keeps the file locked while it is running, so I don't think an automatic sync solution outside of Chrome would be possible.

If you really want the whole skinny on Chrome's Web Data database, you could comb through that part of the Chromium source code for hints.