Given a collapsed state, can we derive the prior shape of the wavefunction? More or less the title.
Assume that we have found a box containing a completely isolated system of particles. We do not know for how long this system has been allowed to evolve. We do know what particles the box contains (let's say it's written on the box).
We now decide to open the box and measure the positions of the particles inside. It is my understanding that these measurements collapse the wavefunction describing the isolated system.
Now, would we be able to derive - from this information alone - the shape of the waveform just prior to us opening the box and measuring the positions of the particles inside?
If not, is there any further information about the system - time since the box was last sealed for example - that would allow us to derive this prior shape?
 A: No, this would be like asking if a coin is fair by only flipping it one time, or determining in what way a die is weighted by only tossing it once.
If you wanted to determine the wavefunction prior to collapse then you would need many similarly prepared systems measured in the same way. From there you could construct the position probability distribution, and hence reconstruct a possible wavefunction prior to collapse.

If not, is there any further information about the system - time since the box was last sealed for example - that would allow us to derive this prior shape?

This is pretty open-ended. Of course knowing the initial wavefunction would be sufficient, but it sounds like you are possible asking for the minimum amount of prior information one would need perhaps? I don't think I am clever enough to think through all of that, but perhaps someone else is.
A: Once you collapse the waveform, the information you see is gone.  However, there are some interesting corner cases which have been explored.  Weak measurement comes to mind.  Instead of measuring the system completely, you do some clever tricks (typically involving entanglement) to only collapse part of the system.  This lets you restore the waveform, other than the parts that were measured.
Handwaving the phrasing a bit.  In practice, the term "weak measurement" is not an exact term
