Can work be done within a system, or does the object have to be outside of a system to do work on it? My example is that I have a ball that is dropped into water. My understanding is that the water does negative work on the ball since it slows it down and also pushes it back up (buoyancy).
Does it count as work if water is included as one of the objects of the system?
If it does, how to calculate the work done on the ball since at different points in time the ball is submerged different amounts, thus buoyancy is different every time.
 A: 
My understanding is that the water does negative work on the ball since it slows it down and also pushes it back up (buoyancy).

This is correct. The work is defined as dot product of force and displacement
$$dW = \vec{F} \cdot d\vec{r} \tag 1$$
Force (buoyancy) direction is upwards, displacement direction is downwards, hence angle between two vectors is $180^\circ$ and the dot product is negative. This means that water does negative work on a sinking object, and sinking object does the same amount of positive work on water.
Please beware, internal forces do not always result in zero net work. Take explosions for example: before the explosion kinetic energy of the body is zero; after the explosion there are a lot of fragments with some (finite) kinetic energy. Since kinetic energy is always positive, total kinetic energy of the system (all fragments) is not zero. This means the internal forces generated some net work, which follows directly from the work-energy theorem $\Delta K = W$. However, in the context of Newtonian mechanics internal forces do always result in zero impulse of the system, which means that internal forces cannot change the system momentum (check "conservation of momentum" if you want to learn more).

Does it count as work if water is included as one of the objects of the system?

If you say the system is a black box with water and sinking object in it, to an observer outside the box no work is done on the black box. Remember that to calculate total work, the force $\vec{F}$ in Eq. (1) is net force exerted on the object.
From third Newton's law of motion it follows that fluid exerts upward force (buoyancy) on the object, and at the same time the objects exerts downward force of the same magnitude on the fluid. If sinking object and fluid are considered as one system, no net work is being done on the system.

If it does, how to calculate the work done on the ball since at different points in time the ball is submerged different amounts, thus buoyancy is different every time.

Time-dependency is irrelevant to calculate work, what it matters is displacement. Buoyancy varies only while the object is partially submerged. Remember that buoyancy is defined as
$$B = \rho_f \cdot V_s \cdot g$$
where $\rho_f$ is fluid density, and $V_s$ is submerged volume of the object. When object is fully submerged, the volume $V_s$ equals total object volume which is constant for rigid objects. If object is partially submerged then you can express submerged volume as a function of displacement $V_s(x)$ and then do the integral in Eq. (1).
