Is the universe in orbit?
Not as far as we can tell.
First of all the word 'universe' is supposed to mean everything that exists. So what would the universe be in orbit around?
That aside, is there a centre around which it is revolving?
Though the universe has various structures, including massive super-clusters of galaxies, there does not appear to be a centre to it.
This is counter intuitive because when we think of the Big Bang we naturally think of it having a centre out of which the universe exploded, but that's not the way it works.
The primary large scale motion we see is expansion. The various galaxies are generally moving apart from one another.
This big question is: Will the expansion slow down and go into reverse, resulting in a Big Crunch or will the universe expand forever?