This is difficult since the properties of cream and egg yolks are hard to replicate by simply using another liquid, and also depending on what kind of ice cream was made for the original recipe. Milk and cream are easier to substitute; as for traditional custard recipes using eggs and yolks, experiment with different organic thickeners for richness. As for a recipe for a vegan ice cream, below is one from the book "The Complete Vegetarian Cuisine," by Rose Elliot. It contains soy milk. The book lists a recipe for Chocolate Ice Cream and the author's change for the Vegan variation. Below are some suggestions the author has for turning it vegan. Her suggestions for the recipe she revised:

1. Using soy milk in the place of whole milk.
2. Replacing evaporated milk or light cream with concentrated soy milk or a non-dairy cream.

The Chocolate Ice Cream recipe and the adapted vegan version are from "The Complete Vegetarian Cuisine," by Rose Elliot.

For those wanting vegan ice cream recipes, try some from this great blog:
A Vegan Ice Cream Paradise.

Chocolate Ice Cream (regular recipe)

3 3/4 c milk
4 oz. sweetened dark chocolate
5 T light brown sugar
1 T cornstarch
1-14.5 oz. can evaporated milk or 1 1/4 c light ceram

Put all but 4 T of the milk in a saucepan with the chocolate and sugar. Bring to soft boil. Mix the starch with the remaining milk. Temper in some of the hot mixture to the cornstarch, whisk well, then add back to hot milk mixture. Stir over moderate heat until slightly thickened, about 2 to 3 minutes. Remove from heat; cool slightly. Add in the evaporated milk. Place in blender and blend until smooth. Pour into a plastic container to cool completely. Freeze until half solid, beat again and return to the freezer until hard.

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Vegan Chocolate Ice Cream (vegan recipe revised from above)

3 3/4 c soy milk
4 oz. carob bar, cut up
5 T light brown sugar
1 T cornstarch
1-14.5 oz. can concentrated soy milk or 1 1/4 c non-dairy cream

Put all but 4 T of the soy milk in a saucepan with the carob and sugar. Bring to soft boil. Mix the starch with the remaining soy milk. Temper in some of the hot mixture to the cornstarch, whisk well, then add back to hot soy milk mixture. Stir over moderate heat until slightly thickened, about 2 to 3 minutes. Remove from heat; cool slightly. Add in the concentrated soy milk or non-dairy cream. Place in blender and blend until smooth. Pour into a plastic container to cool completely. Freeze until half solid, beat again and return to the freezer until hard.

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References used:

Elliot, rose . The Complete Vegetarian Cuisine. New York: Pantheon, 1988.