The problem is just to translate the date in a format suitable to the date command. The more similar to your input is "mmmdd yyyy", so that something like:
date -f <(sed "s/'/ 20/" file) +%m/%d/%Y
should work, except for the fact you have to find a way to distinguish dates from the past century and from the current one.
Regarding awk, it has some functions to manage dates: using a combination of mktime and strftime it can manage dates in the past, but maybe it's far easier to stick with the date command.