No, sendmail cannot turn on recursion in BIND. Sendmail uses the standard resolver APIs which don't provide the feature you want.
Godaddy say
NOTE: We do not allow recursive DNS to run on dedicated or virtual dedicated servers unless it runs locally and for a specific IP range. If we find your server running an improper configuration of recursive DNS, we will exercise our right to suspend your account. The account will remain suspended until arrangements are made to turn off recursive DNS.