Can I use tartaric acid in making Snickerdoodles?

I've never tried it, but you might try 1/2 the amount of tartaric acid based on information from Nigella Lawson :

Cream of tartar is made by combining tartaric acid with potassium hydroxide. This partially neutralizes the tartaric acid, so cream of tartar is less acidic than tartaric acid.

If the tartaric acid is used in baking or added to egg whites before whisking into meringues then it should be possible to use cream of tartar in roughly double the quantities of the tartaric acid, though we have not tested this. However we would not recommend using cream of tartar as a substitute in any other recipe.

You can use baking powder instead of both the baking soda and the cream of tartar, in the proportions explained in this quote:

If your recipe calls for baking soda and cream of tarter, I would just use baking powder. One (1) teaspoon baking powder is equivalent to 1/4 teaspoon baking soda plus 1/2 teaspoon cream of tartar. If there is additional baking soda that does not fit into the equation, simply add it to the batter. (Source)

Baking powder a mixture of baking soda and an acid. Various compounds are/were used as the acids by different manufacturers and at different time periods. Cream of tartar was one of the first acids used in baking powder. So your recipe basically has homemade baking powder. That might be because baking powder was not readily available at the place and time the recipe was written. Or it might be because early baking powders were not very shelf stable, so you would have gotten a more reliable rise by mixing your own baking powder.

If you want more details on this topic, the Wikipedia article has some interesting info.