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.
The concept you're looking for is that of a planet's Hill sphere. If a planet of mass $m$ is in a roughly circular orbit of radius $a$ about a star of mass $M$, then the radius of this "sphere" is given by
$$
r_H = a \sqrt[3]{\frac{m}{3M}}.
$$
For the Sun-Earth system, this yields $r_H \approx 0.01 \text{ AU}$, or about 1.5 million kilometers.
The calculation given in the Wikipedia article shows how to derive this in terms of rotating reference frames. But for a qualitative explanation of why your reasoning didn't work, you have to remember that the moon and the planet are not stationary; both of them are accelerating towards the star. This means that it's not the entire weight of the moon that matters, but rather the tidal force on the moon as measured in the planet's frame. This effect, along with the fact that the centripetal force needed for the star to "steal" the planet is a bit less when the moon is between the star and the planet, leads to the expression given above.
As pointed out by @uhoh in the comments, the L1 and L2 Earth-Sun Lagrange points are precisely this distance from the Earth. These are precisely the points where the gravitational forces of the Earth and the Sun combine in such a way that an object can orbit the Sun with the same period as the Earth, but at a different radius. In a rotating reference frame, this means that the influences of the Earth, the Sun, and the centrifugal force are precisely canceling out; any closer to Earth than that, and the Earth's forces dominate. Thus, the L1 and L2 Lagrange points are on the boundary of the Hill sphere.
Best Answer
You can take the Newtonian expression for the orbital speed as a function of orbital radius and see what radius corresponds to an orbital speed of $c$, but this is not physically relevant because you need to take general relativity into account. This does give you an orbital radius for light, though it is an unstable orbit.
If the mass of your planet is $M$ then the radius of the orbit is:
$$ r = \frac{3GM}{c^2} $$
where $G$ is Newton's constant. The mass of the Earth is about $5.97 \times 10^{24}$ kg, so the radius at which light will orbit works out to be about $13$ mm.
Obviously this is far less than the radius of the Earth, so there is no orbit for light round the Earth. To get light to orbit an object with the mass of the Earth you would have to compress it to a radius of less than $13$ mm. You might think compressing the mass of the Earth this much would form a black hole, and you'd be thinking on the right lines. If $r_M$ is the radius of a black hole with a mass $M$ then the radius of the light orbit is $1.5 r_M$.
So you can only get light to orbit if you have an object that is either a black hole or very close to one, but actually it's even harder than that. The orbit at $1.5r_M$ is unstable, that is the slightest deviation from an exactly circular orbit will cause the light to either fly off into space or spiral down into the object/black hole.
If you're interested in finding out more about this, the light orbit round a black hole is called the photon sphere, and Googling or this will find you lots of articles on the subject.