What are dipole moments in a molecule supposed to act upon?
Your confusion appears to arise from trying to consider dipole vectors as representing forces, which is incorrect.

Dipole vectors don't "act on" anything; they describe the displacement of the "center-of-charge"&ddagger; of a given region of space (including both the electron cloud and the nuclei) from a particular reference point. In particular, I believe you are confusing dipole moment vectors with electric field vectors: 


The electric field vector at a point describes the force experienced there by a point charge
The dipole moment vector at a point describes the displacement of the "center of charge" of the system from that point


Thus, whereas the electric field contributes to the forces present in a system, the dipole is simply a post facto descriptor of a system.



Further, in this case the law you quoted is not applicable:


  Two equal and opposite vectors do not cancel each other out when acting on different bodies.


It only makes sense to calculate an overall dipole moment if you use the same reference point for all of the 'subsystem' dipole vectors. Thus, in this example of $\ce{CO2}$, the two $\ce{C-O}$ 'subsystem' vectors point in exact opposite directions from the same reference point, since $\ce{CO2}$ is linear.  Thus, since the outside atoms are both oxygens, the magnitudes of both 'subsystem' vectors are equal and their sum is the zero vector.



&ddagger;
For a non-electroneutral region of space, by center of charge I mean the charge-weighted centroid of the system: (1) Choose a reference point $\vec r_0$. (2) At each point $\vec r$ in the region of interest, multiply the charge at that point, $\rho\!\left(\vec r\right)$, by the distance from the reference point, $\vec r - \vec r_0$. (3) Integrate the resulting product, $\rho\!\left(\vec r\right)\cdot\left(\vec r - \vec r_0\right)$, over all space. (4) Divide by the net charge of the region of interest. Dipole moment calculations are more complicated for electroneutral systems, but the principle is similar.