Well, I found a solution to the issue here: https://bugs.chromium.org/p/chromium/issues/detail?id=1315684#c17
Specifically here is the fix that worked for me:
However, setting /org/mate/desktop/sound/input-feedback-sounds *did* fix it for me on MATE.
I used dconf-editor to set it to false, but you can also use:
dconf write /org/mate/desktop/sound/input-feedback-sounds false
or
dconf write /org/gnome/desktop/sound/input-feedback-sounds false