Variable Power Supply to ICs
There is nothing inherently wrong with doing this in theory. In fact, modern CPUs have dynamic voltage scaling which varies voltage depending on CPU load (as well as clock frequency), throughout normal operation. Its particularly prevalent in portable electronics that need to save power as much as possible.

This is a difficult question to answer, because we can't say "Yes it'll be fine in your design" without thorough testing/simulation - but the concept is sound.

Things to watch out:


Transients - make sure there is plenty of bulk decoupling capacitance
to smooth out the rising and falling transitions between voltage
levels. Inductors can make this worse.
Every condition is anticipated - Although you expect only a small set of conditions, you need to make sure that none of the combinations could cause issues (5V only, 3.7V only, both, none).