I usually make carrot cake with applesauce, using the same amount of applesauce as oil is called for, and it comes out a bit more pudding-y than springy and fluffy. I actually prefer my carrot cake this way, so the success of the apple sauce probably depends somewhat on the type of cake you're making.

When my daughter was in high school, for her science project, we baked 3 cakes according to a specific oil-based recipe. We substituted 1/2 applesauce for 1/2 of the oil in one, and all the applesauce in another. She held a taste test. The all oil cake was judged the best in flavor and texture; the all-applesauce cake was the worst. I say go for the best taste and texture and eat a little bit. Cake is a treat--make the best cake you can. (Just my humble opinion....)

Applesauce can be substituted anywhere from 3/4 to 1 cup for 1 cup of oil. There really isn't a specific conversion b/c it can depend on what exactly you are making and sometimes trial & error is the best bet.

However, keeping the amount of sugar as low as possible in your cake recipe would be the healthiest choice, and therefore I would just use the oil (as long as you are using a quality, healthy oil like coconut oil) and resist the urge to add even more sugar (applesauce) to the recipe. Your waistline will thank you.