Is it possible to make a pie that does not include gluten or dairy? What techniques and ingredients can I use?
You have several options for the crust. It is possible to purchase mixes that will get you started. There are also several recipes available. A good resource for all your gluten-free baking is Gluten-Free Girl and the Chef, and you can find a recipe there. Another great option is former pastry chef Helene at Tartlette. This question has compiled a list of gluten-free baking recipes which may also provide you good jumping off points.
You can make a dairy free crust by replacing butter with lard or vegetable shortening. Margarine is likely not be dairy free. Lard will provide a texture and taste that is better than an all shortening crust, but shortening is easier to find in stores. If the dairy free requirement is a requirement to avoid casein, you will have to do some experimentation on your own to find an appropriate egg replacement, as gluten free crusts appear to require eggs to hold together. Options for egg replacers can be found in this question and this question.
Once you have assembled a crust recipe from these tips, if you have questions about it or are struggling you can ask a follow-on question here about how to improve the crust and the community can provide specific help on a provided recipe.
In general it appears that the gluten free crusts available are actually similar to work with versus a regular pie crust, something that is not always the case in gluten free baking. As usual you want to do everything cold and refrigerate your dough over night. Pie dough of all forms is easiest to roll when you place it between two floured sheets of parchment paper. Remove the top piece of parchment paper, flip the bottom piece with crust on it (hold it all together, because your crust shouldn't be sticking) into the pie plate or onto your pie, and remove the top piece.
Apple pie is sometimes thickened with flour, and obviously you do not want to do this when cooking gluten-free. The absolute best pie thickener is actually not flour, so you are set here. In taste tests the editors of Cooks Illustrated determined that instant tapioca, a gluten-free product, is the best pie thickener. Make sure, however, not to use more than 2 tablespoons or your pie may be gummy. Since apple slices contain pectin, you shouldn't need more than 2 tablespoons for your pie to set up.
Sugar, apples, and the usual apple pie spices (if you choose to spice your pie - with good apples you can actually make an amazing pie with simply sugar, instant tapioca, and apples) should all be gluten free and dairy free, with the usual caveat that in gluten-free cooking and baking you want to make sure that there has been no gluten added to anything by carefully reading labels.
Spread apples and jam out on a corn tortilla and then bake in the oven for individual gluten free tartlets. Quick, easy, tasty, and a little odd.
I make an Indian Corn Pudding with a bottom layer of apples that has the same feel as pie for dessert, but without the gluten. http://cookloose.com/index.php?n=Recipes.IndianCornPudding