Log into smile.amazon.com with your Amazon credentials.
Click the name of the current charity, to the right of, "Supporting."

Click "Change your charity" at the bottom left corner of that window.

In that field, below where it says; "Or pick your own charitable organization:" enter the name of the organization you want to benefit by your (and Amazon's) charity. Click Search. Click "Select" to the right of the name of the organization in the search results.