Does our galaxy orbit anything in the universe?

Yes, our galaxy, the Milky Way, is part of a group of galaxies called the Local Group. Within the Local Group, the Milky Way and the Andromeda galaxy are the two largest galaxies and they are actually moving towards each other. It is believed that the Local Group is also moving towards the Virgo Cluster, a larger cluster of galaxies. So, in a sense, our galaxy is moving within the universe, but it is not orbiting around a single object like a planet orbits a star.