JR train route

Does JR trains go between Shinjuku station and Osaka?

Yes, it does.

Information about travel between Tokyo and Osaka:
https://www.japan-guide.com/e/e4006.html

A tool to look up train connections (by deselecting the "Private Railway" option in the search details, you can get JR-only connections):
http://www.hyperdia.com/

If you mean directly, though, that's a no.