I am applying for an institute and would like to upload my manuscript for a paper which is under peer review onto the Supplemental materials section. The manuscript contains the watermark "For Peer Review Only." Can I upload this version?
It is not uncommon for working papers to be uploaded to academic websites prior to, or during, peer review. The only possible impediments I can see to this would be if the agreed terms of review of the journal you have submitted to prevent you from doing this (i.e., publishing the draft during review) or if the terms of the institute's website prevent this. I doubt that either is the case, but if you're not sure, check the terms.
In general, yes, you are free to post it.
Given that you wrote the manuscript (with co-authors) you are free to do as you please, legally, because you still own the copyrights. This is also why preprints cannot legally be restricted.
Some journals might not permit you to post preprints as their own submission rule, which you can check using SHERPA/ROMEO. This is relatively rare nowadays as preprints are becoming more and more accepted. Specifically to posting the version watermarked "Under Review", it is still your creative work (jointly with co-authors), so you can post this too.