I have not tried making this ice cream without vegetable glycerin, and I really recommend using the glycerin for best texture. The glycerin also helps keep the ice cream from sticking to the sides of the ice cream maker (ice cream made with coconut milk tends to do this even more than regular ice cream, I've found). I purchase vegetable glycerin off of Amazon (link below), or you can find it at some local cake decorating stores. Whole Foods carries NOW brand's food grade vegetable glycerin in its skincare section.
Most people with nut allergies can handle coconut just fine, but check with your doctor first to make sure.