It may seem obvious, but oily, acne-prone skin doesn't need more oil layered on top of it, even if it does need moisture. As a matter of fact, very oily skin may not need to be moisturized at all. This is especially true for people with adult acne. That said, most people can benefit from a pea-sized dollop of moisturizer, even if they have breakouts.
You should look for water-based moisturizers formulated for oily or breakout-prone skin. It also helps to look at the ingredients list on the back for anything that may trigger breakouts in some people, like mineral or coconut oils. For sensitive skin, glycerin is a safe base for a moisturizer because it won't irritate skin. What's more, glycerin is a humectant, meaning that it pulls moisture from the atmosphere to keep skin balanced. Some moisturizers include ingredients like the tried-and-true benzoyl peroxide or the newer salicylic acid, which can help keep acne in check as it maintains the skin's moisture balance.