I am Admin for a few Facebook Groups. When a request comes in I get a notification on Facebook. I have to then click approve. I am very active on Facebook and keep up with my notifications. Notifications are like emails, some people keep up with them and open and respond quickly, some people dont. It is possible that the person who is admin of the group just is not keeping up with notifications and has not got around to checking and approving new member requests. Or the admin of the group is not that active on Facebook and does not log in that often.
It is also possible that the admin of the group has not seen your PM as on Facebook PM's from people you are not friends with go to a different folder. I have missed a few PMs as I didnt get a notification for the Other folder.
You could check the member list and see if you know anyone already in the group who may know the admin. That way you could ask this mutual person to tell the admin to check the new member requests.