Where does Alfred store web searches?
(Note, 1) I have my Alfred preferences located on my dropbox, but the default location is at ~/Library/Application Support/Alfred 2, i.e., in your home folder's Library folder, and 2) the following is for the Alfred 2 app)

Using sudo opensnoop | grep Alfred.alfredpreferences in terminal (if using this you'll need to provide your admin password), I see that adding a custom web search updates the following file within the Alfred.alfredpreferences.

Alfred.alfredpreferences/preferences/features/websearch/prefs.plist in the Finder.

You'll need to right-click on the Alfred.alfredpreferences file to "Show Package Contents" to view these folders.

Once you've navigated to the .plist file, it can be view via Quicklook by pressing the Space Bar whist the file is highlighted.