I love the gluten-free recipe, but I also have to cook dairy-free. Could this recipe be made with coconut milk, almond milk, or another substitut...

I can't guarantee this will work, but if I were making this dairy-free (as I do with a lot of things), I would use a fat-rich coconut milk in place of the cream. See how HLA makes her coconut cajeta here:
http://food52.com/recipes/9557_coconut_cajeta_chocolate_fondue
If you don't add the chocolate, it turns into this incredibly soft dairy-free caramel. You could probably just make this and use it instead of the entire caramel part of the recipe. I use it here, in my banana pudding:
http://beyondcelery.blogspot.com/2011/03/ginger-caramel-banana-pudding.html

Plus, lemon and coconut go really well together. I think you could totally use almond milk in place of the milk in the first part of the recipe, but it will be thinner and less rich. Using coconut milk there as well will give it that nice creaminess, plus tie the flavors of the caramel in together.

For the lemon cream, I'd use vegan buttery sticks in place of the unsalted butter. I've made pastry cream with them before, using a 1:1 replacement, and it's worked just fine. (Even when the pastry cream recipe calls for unsalted butter and the vegan buttery sticks are salted. Just leave out the pinch of salt.)

For the cake itself, I think using a milk sub would work just fine. Almond milk would be my preference. However, for the pastry cream, I'd recommend looking up a different dairy-free version as I wouldn't know where to start with replacing the cream