Hello furquan,
In WHM > Server Configuration > Statistics Software Configuration area under Generators Configuration section, uncheck "Active by default" for Awstats and check "Available to users" for Awstats. Click the "Save" button there.
In that same area, click on "Choose Users" button under User Permissions area, then select the user in question and click the "Add" button.
What the above options will do is allow the user to select Awstats in cPanel > Choose Log Program area. All other users will be locked out from selecting Awstats. Since they cannot select Awstats and it isn't active by default, no other account would have it available.
Thanks!