Here is how to do that -
Give them a list of a few people to buy for.
On the list have some ideas of what that person would like for Christmas. Make sure it is someone that they know so they can personalize with that gift.
Give them a budget to stick with for ALL the presents.
Have them divide that amount among the people on their list. They will know how much they are to budget for each person. Finding a good deal is an awesome feeling! Take your child to the store with the list and have him/her look and choose the gift. If he/she wants to spend a bit more, explain that it is ok, but you still have to stay within the whole budget. So if he/she spends a bit more on one person, the next one will have to be a bit more price friendly.
Once the gifts are purchased, have the child wrap and label the gift. Personal name tags are always fun.
Have them personalize the gift with homemade gift wrapping too. Maybe with the recipients name on the package?
And last, make sure you go over the list and budget to make sure your child hasn't forgotten anything and is within his/her budget. Make sure the package is ready to ship or give in a timely manner. If it needs to be shipped have the child do that as well. He/she can make a shipping label and pack it in the shipping box.
And don't forget to give your child praise for the accomplishment of responsibility! They might even have a little more appreciation for the next gift they receive.