Mix flour, salt, and cream of tartar in a large pot. Add water and oil.
Cook over medium heat until the mixture pulls away from the sides of the pan (about 5 minutes), stirring constantly.
Add vanilla extract (for smell, not taste). Mix thoroughly. Put play dough on a clean surface. When cool enough to handle, knead lightly.
Store in airtight container.
Add food coloring to the water to make colored play dough. Add sparkles during the handmixing time for sparkly play dough.
Notes
Cream of tartar is not dairy.
Substitutions
Gluten: Gluten is a protein found in specific grains (wheat, spelt, kamut, barley, rye). Other grains are naturally gluten-free but may have cross-contact with gluten-containing grains. Look for certified gluten-free products if you need to avoid gluten. Find out more about wheat and gluten substitutions.
Milk and Soy Substitutions: Alternative dairy-free milk beverages and products will work in most recipes. Find out more about milk substitutions and soy substitutions.