The icons are in the file
C:/Program Files (x86)/Chromium/chrome.dll
You can use an application such as 7-Zip to browse the contents.
(You may need to close Chromium before you can access the file - or create a copy.)
chrome.dll/.rsrc/BINDATA/ and you will see a bunch of numerically named files.
Files 6167 to 6192 are the avatar icons, in PNG format.
Unfortunately 7-Zip doesn't seem to support putting changed files back into the DLL, so need to find other software to do that...
Warning: modifying the chrome.dll does change the image, but also appears to stop keyboard shortcuts from working...