As far back as 1960, Lyndon Baines Johnson had a Texas law changed so that he could run for re-election to his US Senate seat at the same time he was running as John F. Kennedy’s Vice President. In 1988, Senator Lloyd Bentsen did the same thing so he could run for both the US Senate and Vice President. Connecticut Senator Joe Lieberman (VP in 2000), Delaware’s Joe Biden (VP in 2008), and Wisconsin congressman Paul Ryan (VP 2012) all did as well. Source
That's a little unclear, though. Johnson had the law changed. I don't know if the others had to do that.
Doesn’t look like KY intends to change their laws so Rand can run.