Is reproduction of copyrighted material in a research proposal fair use?
A proposal is not a publication, and as such you do not need to worry about the usual strictures of copyright.  It doesn't really matter what the technicalities are: if the proposal is to remain private, de facto it is as though it doesn't exist from the copyright holder's perspective.

Academic honesty, however, is another story.  If the figure is not yours, you absolutely must give credit to its actual author.  Further, you need to ask yourself: is including somebody else's work actually good evidence for my ability to execute the work that I am proposing?  In some cases, it may still be the right thing to do, but usually if it is a diagram, you will probably do better to create your own independent version, and if it is data you will probably do better to cite the result but not to waste precious proposal space reproducing their figure.