How would one construct a 3D model of a molecule based on the crystal structure of its salts?
In general, the crystal structure of a molecule is a highly accurate geometry. So yes, you can take a salt and modify as needed. (The caveat in the comment is useful - keep in mind that different groups may be protonated or deprotonated depending on solvent and pH.)

There are two other caveats:

1) The conformation in the crystal structure is not necessarily the lowest energy conformer in solution or the gas phase. You may wish to perform a conformer search to find a low-energy minima.

2) If you are performing further calculations (e.g., DFT for electronic structure, force fields for MD, etc.) the geometry of the crystal structure will not likely be a minima of the computational method, so geometry optimization or equilibration (for MD) is needed before further simulation.

That said, the crystal geometry is a good place to start for many uses. Certainly if a crystal structure is available, I'd suggest you use it rather than build a geometry by hand.