The answer to this is a bit nuanced so I have now edited the answer to make it as clear as possible.
The big picture is: Yes, it is possible to measure a wave function, if it is possible to find or produce particles with that wave function so that repeated measurements can be made (not on the same particle each time, but on a different particle which is known to have the same wave function). This can be done in practice. For example, this experiment effectively measures the magnitude of the wave function in momentum space for an electron in a hydrogen atom: source.
If you write $$\psi(\vec{r})=A(\vec{r})e^{i\alpha(\vec{r})}$$ For $A,\alpha$ real, then $A$ can be determined by measurements of position, and I believe it is possible to find $\alpha$ only from momentum measurements up to a constant offset $\alpha(x) \to \alpha(x) + c$. Because of this constant offset which is not measurable, $\psi(\vec{r})$ is not measurable at a point, but $|\psi(\vec{r})|^2 $ is, $A(\vec{r})$ is, and $\alpha(\vec{r})$ is up to that "offset", which is called a global phase.
Of course, any finite number of measurements will only give information on a finite number of points of $\psi(\vec{r})$, each with uncertainty as well, especially because many measurements are necessary to measure a probability. These measurements must be done by repeatedly preparing a particle in that wave function and then measuring at a chosen later time. The experimentalist does not need to know what the wave function is beforehand, she/he must only know that the wave function is the same for each particle. Then, she can interpolate in between the measured points to come to a good guess for $\psi$ at all points. In practice, any experimental outcome which is predicted by $\psi(\vec{r})$ can also be used to constrain it.
There is a practice of measuring states called Quantum Tomography. It has its own wikipedia. 795
It is usually used for finite-dimensional systems. But in reality every system is infinite dimensional, one only treats them as finite-dimensional. One could treat the position space wave function in the same way.
That being said, though it can be done in principle, I'm not aware of a full tomography which has been done in practice. If one really doesn't exist, this is a severe experimental weakness of testing the theory in my view. If anyone has a source to one, I'd be interested. I know of many momentum experiments where the fourier transform of the wave function was in fact checked, but not the entire thing.
I will point out again for clarity, because it seems that others are of the opinion that I have not emphasized this enough, that if you do not have the ability to find or produce many particles in the unknown state $\psi$ then it is not possible to measure in practice. It seems to me, though, that a clever setup could be used to measure almost any wave function in principle. Whether technology has reached that point is another question. But I will do due dilligence and flag that last statement as opinion, because I cannot provide references or proofs that it must be so. 795