Is it possible to 3D print shapes of atomic orbitals for a regular customer? Disclaimer: I am not a professional 3D printer, so I'm not really familiar with how 3D printing works.

I was thinking about trying to make a real life model of the atomic orbitals, to clearly see how the orbitals are really shaped. I thought about trying to 3D print a block, made up of colored transparent "ink", in such a way that the darkness of the color at a point in the block should be proportional to the ψ2 value (probability of the electron cloud). This would print a block in which high density areas are darker than light intensity areas.

My question is, would it be possible to design an algorithm to print a specific darkness of ink at a specific location based on the computed value of ψ2, which is obtained by solving Schrödinger's Equation. Also, would it be possible to have this fullfilled at a reasonable price to be afforded by a regular customer, such as an engineering employee?
As Darth pixel mentioned, your best bet would likely be to explore subsurface laser engraving which makes little bubbles inside acrylic to make it translucent (frosted) on the inside in certain places.  However, it is expensive, it has no colors, and the resolution isn't usually great (it may look speckled on the edges instead of having a smooth gradient from clear to whitish).

A better option might be to 3d print a probability map of a 2D image an orbital cloud where height corresponds to probability rather than color. (So it will look like a hill or group of hills)  This would only run you a few dollars.