Im also looking for this fix. It seems theres a /usr/local/cpanel/etc/awstats.conf file that should enable this but im not sure how to do it. There is a section for this:
# When this parameter is set to 1, AWStats add a button on report page to
# allow to "update" statistics from a web browser. Warning, when "update" is
# made from a browser, AWStats is ran as a CGI by the web server user
# defined in your web server (user "nobody" by default with Apache, "IUSR_XXX"
# with IIS), so the "DirData" directory and all already existing history files
# (awstatsMMYYYY[.xxx].txt) must be writable by this user. Change permissions
# if required.
# Warning: Update process can be long so you might experience "time out"
# browser errors if you don't launch AWStats enough frequently.
# When set to 0, update is only made when AWStats is ran from the command
# line interface (or a task scheduler).
# Possible values: 1 or 0
# Default: 0
#
AllowToUpdateStatsFromBrowser=%BROWSERUPDATE%
I believe that changing this to 1 will work, but as I have tried it and it hadnt worked I am still looking for a solution