for the electron in the Hydrogen atom, the orbital motion doesn't interact with the electron's spin, so "the wave function" pretty much means just a complex $\psi(x,y,z)$. You may choose the electron's spin to be up or down independently of that.
However, you must realize that a wave function of three quarks has many more components. First of all, there are three particles in relative motion rather than two. It means that even if you decouple the center-of-mass coordinates, the relative wave function is a function of six coordinates, $x_1,y_1,z_1,x_2,y_2,z_2$.
So you can't really visualize the full wave function in a simple way because it is a function of six variables rather than three and of course, it doesn't factorize into a product of functions of three variables (something that is strictly speaking true even for all atoms more complicated than the Hydrogen atom).
Moreover, you must understand that the quarks have many discrete degrees of freedom that are not decoupled from the orbital motion in this case. Each of the three quarks has 1 of 3 colors and 1 of 2 possible values of the spin. The combinations of the color really force you to consider 27 combinations of the quark colors - 27 different wave functions (only 6 of them are nonzero and equal to each other, up to a sign) - and 8 combinations of the three quarks' spin. The spins are correlated as well.
So the right wave function for 3 quarks is really a set of $27 \times 8$ complex functions of six variables. Of course, you may visualize various aspects of these functions by integrating it in space and so on.
It's important that the strong force between the non-relativistic quarks is spin-dependent. That's why the protons' spin is not arbitrary - it equals $1/2$ instead. So one of the quarks' spin differs from the remaining two.
Even if you managed to invent ways how to visualize the $27\times 8$ wave functions of 6 variables describing the relative positions of the three quarks, with the best correlations between the colors, spins, and motion, it would still be a hugely oversimplified model of the proton. In reality, it is not true that the proton only contains 3 quarks. Those 3 quarks we normally talk about are analogous to "valence electrons" in an atom but there is also a large see of equally real gluons and quark-antiquark pairs - additional partons whose total color vanishes.
To really describe the wave function of the proton, you need to talk about the right theory of the proton's structure - quantum chromodynamics (QCD) - which is an example of a quantum field theory - one discovered in the early 1970s. It has infinitely many degrees of freedom, like any field theory, so the right "wave function" is really a "wave functional" or a function of infinitely many variables.
You must realize that many approximations valid in atomic physics break down. For example, the speed of electrons in the atoms is very small - effectively controlled by the fine-structure constant $1/137$ that tells you the speed in the units of the speed of light. That allows you to use non-relativistic quantum mechanics. However, the proton's "strong" fine-structure constant relevant for the "three quarks only" is close to one, so the speed of quarks in the proton is always comparable to the speed of light. Consequently, the extra relativistic energy/mass is comparable to the rest mass as well as the interaction energy, and the proton always has enough energy to create quark-antiquark pairs and so on. It's a mess where relativity is needed much like particle creation and annihilation.
In some sense, it's true that "the 3 quarks", if you select them from the sea of the infinitely many partons, like to occupy three different regions in the ground state. But to be more accurate about what this statement actually means for the wave function(al) of the actual proton, you would have to go through a series of so many approximations and idealizations that it's not worth it. The goal of quantum field theory is not to visualize the structure of something; the goal is to calculate the results of the experiments and one can't really design good experiments that would probe the detailed shape of the wave function of the proton.
The parton distribution functions are the closest observables to this information. However, their very assumption is that the proton has infinitely many rather than 3 partons (quarks, antiquarks, or gluons). To summarize, don't expect anything that is as valid yet as useful as the pictures for the atomic physics. Proton's analogy to an atom has severe limitations.