Globally disable "include online search results"

The command you are looking for is:

gsettings set com.canonical.Unity.Lenses remote-content-search 'none'

To make the command globally (for all users) on log in, you'd have to add a launcher in /etc/xdg/autostart, running the command:

/bin/bash -c "gsettings set com.canonical.Unity.Lenses remote-content-search 'none'"

The file would look like:

[Desktop Entry]
Name=Disable Search
Exec=/bin/bash -c "gsettings set com.canonical.Unity.Lenses remote-content-search 'none'"
Type=Application

Save it as disable_onlinesearch.desktop in /etc/xdg/autostart

Remove the Amazon launcher

To make sure no user can use the Amazon launcher, it is best to remove it globally: go to /usr/share/applications and look for the file:

ubuntu-amazon-default.desktop

and remove it.

Alternatively

If for some reason, you'd like to keep the file ubuntu-amazon-default.desktop, but (temporarily?) disable it (make it invisible in Dash/Unity) add a line to the file:

NoDisplay=true

Later versions

The solution(s) above works for the current versions of Ubuntu, at least from 14.04 and higher. (as on May 28, 2015).

you can completely remove unity-webapps-common to remove the shopping lens, including amazon.

sudo apt-get purge unity-webapps-common

You then need to log out and back in.