If the digestive system is approximately 1-dimensional, as with a digestive tract, or even a network of tubes connected together in some way, then a 3-dimensional body will be able to house it. This is because any graph (aka the network of tubes) can be embedded in three dimensions, with the graph complement connected (ensuring that the body housing the tube network is connected and doesn't fall apart).
However a 4-dimensional body does allow for more possibilities of higher dimensional digestive systems. For example you may want your digestive system to be the shape of a 2-dimensional plane.
Picture your animal as a 3-dimensional solid sphere with a slice cut right the way through it, which I will call its stomach, splitting its body into an upper and lower hemisphere. It could insert food in the shape of a large 2-dimensional plane (e.g. a large leaf) into its stomach without the need to cut the food up, and it could digest the food simultaneously in all parts of its stomach. Perhaps only some substances in the food can be digested, so in the stomach these substances are broken down and absorbed into either the upper or lower hemisphere of its body, and the rest of the food passes out the other side of the animal still as a solid plane. If the animal were 3-dimensional you would need to connect the upper and lower hemispheres by tendrons of some sort, which would then have to cut through the food as it passes through - which might be difficult if it ate really tough leaves. But with a fourth dimension these tendrils could loop round by perturbations in the fourth coordinate and avoid cutting through the food. In fact they needn't be tendrils, they could be continuous gluings of the two hemispheres that loop round using the extra dimension.
I see this creature as a bug that clamps itself onto a large leaf and crawls along it, digesting as it goes. Of course this assumes that there are still 2-dimensional leaves in our new 4-dimensional world.