In addition to the README change, here is a change to the actual script, so it works out of the box. Attached a diff that includes Christian's fix.
diff -uNr awstats-7.6+dfsg.orig/debian/prerotate.sh awstats-7.6+dfsg/debian/prerotate.sh --- awstats-7.6+dfsg.orig/debian/prerotate.sh 2016-12-12 08:51:39.000000000 -0200 +++ awstats-7.6+dfsg/debian/prerotate.sh 2018-01-08 16:35:07.309289493 -0200 @@ -2,5 +2,5 @@ UPDATE_SCRIPT=/usr/share/awstats/tools/update.sh if [ -x $UPDATE_SCRIPT ] then - su -l -c $UPDATE_SCRIPT www-data + su -s /bin/sh -l -c $UPDATE_SCRIPT www-data fi diff -uNr awstats-7.6+dfsg.orig/debian/README.Debian awstats-7.6+dfsg/debian/README.Debian --- awstats-7.6+dfsg.orig/debian/README.Debian 2016-12-12 08:51:39.000000000 -0200 +++ awstats-7.6+dfsg/debian/README.Debian 2018-01-08 16:35:07.305289484 -0200 @@ -64,7 +64,7 @@ prerotate if [ -x /usr/share/awstats/tools/update.sh ]; then - su -l -c /usr/share/awstats/tools/update.sh www-data + su -s /bin/sh -l -c /usr/share/awstats/tools/update.sh www-data fi endscript