Probably the easiest way to do this is to make apache a member of the group that has access to /home/mike/orders and /home/mike/orders/orders.txt and make give that group write access.
It may be more secure to create a new group and user, both called, say, orders and make /home/mike/orders and /home/mike/orders owned by orders
rders. Then make the users apache and mike members of the group and give them write access.
Anyway, first off what does the following return?
ls -l /home/mike/orders