As the comments point out, there's no measuring velocity from the frame of something traveling at $c$. The question becomes non-relativistic if we take a single reference point not traveling at c and measure the rate of change of displacement (our proper length) with respect to time (our proper time) between the quanta, rather than asking the unanswerable question of relative velocity from the point of view of something traveling at $c$.
If we represent our $d\vec s/dt$ vectors graphically, we're just trying to find a configuration of vertices with equal lengths between all vertices. In two dimensions, this is an equilateral triangle. In three dimensions, it's an equilateral tetrahedron (a triangular pyramid).
So, the number of quanta between which we can measure a rate of change of displacement equal to c is four, one for each vertex.
One example, with three massless quanta and one massive quanta, would be three receding at c away from us along the near edges of an equilateral tetrahedron (60 degrees of angular distance between each of the three paths), and one stationary next to us.
We can also do this with four massive quanta. In that case, again, this is a non-relativistic problem. Set it up by starting us at the center of an equilateral tetrahedron. We fire one bullet towards each vertex at velocity v. The displacement vector between each bullet is the long edge of a 30-30-120 triangle with a short edge of length v, so the long edge has a length of $\sqrt {3} v$. Solve for v to get $c = ds/dt = \sqrt{3} v \implies v = c/\sqrt{3}$
Now we can give all four quanta any additional velocity $\vec u$ relative to us that we want provided that $|\vec v + \vec u| < c$. In fact, the first case turns out to be just a special case of the four-massive-quanta case at the limit of
$|v_1 + u| \to c$
$|v_2 + u| \to c$
$|v_3 + u| \to c$
$|v_4 + u| \to 0$
The other possibilities: all four massless, two massive and two massless, just one massless, are no good.