The CMB radiation is actually a relic from recombination, not directly from the Big Bang, but let's ignore this for the moment because we can understand why the CMB is omnipresent by considering the Big Bang.
The key point to understand is that the Big Bang wasn't something that happened at a single point. Instead it happened everywhere in the universe. This is because as you wind the clock back towards zero the separation between any two points in the universe decreases, and indeed falls to zero at the moment of the Big Bang. At the Big Bang every point in the universe was at the same place. You may regard this as physically meaningless, and indeed it is. In General Relativity the Big Bang is a singularity and cannot be described using GR, though GR can get arbitarily close to it.
We expect that some theory of quantum gravity will eventually replace the Big Bang by a more physically meaningful description, but even so we expect that very early in the history of the universe all spacetime points were essentially identical. Quantum fluctuations associated with the ending of inflation introduced the inhomogeneities that eventually led to the structure (stars, galaxies, clusters, superclusters) that we see today, but at the time of recombination every point in the universe was still approximately identical and radiated the CMB with the same intensity. That's why the CMB measurements show approximately the same temperature in every direction, with fluctuations of only 1 part in 100,000.
So the CMB is omnipresent in the sense that every bit of the universe generated it with the same intensity. However photons don't stand still, so as the comments say, the CMB photons we measure today come from the surface of a sphere of radius about 46 billion light years. But there is nothing special about this sphere centred on us. Every observer everywhere in the universe would measure the same as we do.