Imagine 4 planets, A has moon a and B has moon b. A and B are binary planets. Is it possible that a and b and A + B all have same the period, so that the 4 planets are collinear?
I will simplify this problem by assuming that the only forces come from Newtonian gravity and by limiting the masses of the moons to much smaller values than that of the planets, such that the moons exert a much smaller gravitational forces on all other bodies and therefore can be neglected; so the only sources of gravitational forces are the two planets.
The two planets will orbit each other according to a Kepler orbit. In this case the only co-rotating positions for the moons would be the Lagrange points. There are five of such points, however only three will also be colinear; one between the two planets and two on the outside, one past each planet. These points will only be colinear if the orbits of the planets are perfectly circular. The downside of these three colinear Lagrange points are that they are all unstable, thus if the moons would get any perturbation then they will eventually leave the colinear alignment with the planets.
So it might be possible that two binary planets with a moon each orbit each other colinear, but this will only be a temporary configuration.