The centre of mass of a system is simply the weighted average position of the mass distribution in that system. Since the universe is thought to be homogeneous and isotropic, any observer should roughly observe themselves as being at the centre of mass for their observable universe. However, I do not think that is quite the answer you were looking for.
From the context, it seems as though you are wondering if there is perhaps a central point around which everything in the observable universe "orbits". In short, the answer is no.
Bear with me, this might get complicated and I am no Richard Feynman; complicated explanations are not my forte. But let me start by saying that gravitationally bound systems (meaning objects orbiting a central point) are sometimes referred to as structures. What you are suggesting is a structure the size of the observable universe. Ok, definition done, now the explanation.
In the beginning (epic orchestral flare), or at least just slightly after the beginning, the universe was homogeneous and the size of the observable universe was essentially the entire universe. Everything was in causal contact and there was virtually no limit to the size of large scale structures. Then inflation began. During the time of inflation, the universe expanded so rapidly that the regions in contact with each other vastly shrunk. The causal horizon of the universe grew much smaller relative to the size of the universe itself. As a result, most structures ended up becoming much larger than the size of the observable universe. Even scales the size of the solar system were outside the range of causal contact. As a result, these structures became frozen in time. One part of a structure could not feel the influence of the other parts and so they could not continue to change or evolve.
When inflation ended, the universe went through periods where radiation and then matter were the dominant forms of energy present. During these periods, the gravitational attraction of matter and radiation caused the rate of expansion of the universe to gradually slow down. This allowed the causal horizon of the universe to expand; information could finally travel between objects in some of the smaller structures. As this continued, first the small scale structures like star systems entered into causal contact. This allowed stars to form and planets around them. Soon, the size of the observable universe was large enough to allow galaxies to form. And later clusters, superclusters, and cosmic filaments.
But then something happened. Dark energy recently became the dominant form of "energy" in the universe. This has made the rate of expansion begin to accelerate again. And so, the relative size of the universe, in which things can causally communicate, is again starting to shrink. The problem? Although the ancient structures from before inflation that are the size of our observable universe can now communicate with some common centre, there has not been enough time for them to evolve and develop fully into a structure with orbits or a common centre of rotation. It is still true that one side of such a structure cannot communicate with the other side.
The largest scale for a gravitationally bound structure is determined by whether enough time has passed, since the cosmic horizon became larger than that scale, for primordial fluctuations (structures) to evolve in a non-linear way (meaning have the complexity to form gravitational structures). The largest scale that can exist today is slightly larger than super-clusters. Then cosmic filaments are the largest scale but even they are a much smaller scale than the size of the observable universe.
Ok, I think I have sufficiently forgotten where I was going with this. The point I was trying to make was that there simply is not enough of a causal connection between objects at distances the size of the observable universe for there to be a common centre of orbit. And because of the new period where dark energy dominates, there is not likely to be any gravitationally bound structures larger than the ones we have today. Cosmic filaments are the largest out there. So no, there is no universal centre of mass as you would be looking for.