I know that the reaction force acts perpendicular to the surface in contact with the body. But where exactly (on the body) does it act?
When we deal with the forces on bodies of finite size, we usually refer to the force on the centre of mass.
What's our reference when it comes to reaction force? Where does it act?
Any force has a line of action and it can be said that the force vector acts anywhere along this line. This line has a direction and at least one location in space to be fully specified. To fully specify a force you need the line of action and the magnitude of the force.
A contact force has a direction along the contact normal and it acts through the contact point. Thus is can cause rotation about the center of mass when the line of action is offset from the COM. The geometry of the contact force is specified by the geometry of the contacting surfaces alone.
A reaction force is specified as any force that does no work on the rigid body. A body has no relative motion in the direction of a reaction force. The geometry of a reaction force is specified by the allowed motion (degrees of freedom) the body and the applied forces.
For example:
A pinned body has no linear motion at the pin location and therefore the reaction force(s) on the pin have to pass through the pin (as to create no torque about the pin). If a pinned body accelerates it is because of an applied force (like gravity) acting on it and not because of the reaction forces. The direction of the reaction force is determined by the applied forces.
A free body in contact with the ground will have a contact force along the contact normal as specified by the contacting surfaces. The body is not allowed to have motion only along the contact normal. So the contact force is also a reaction force, but the line of action is specified a-priori in contrast to the previous example.
We can always assume that the force acts on the Center of Mass. Any Force that acts on a rigid body adds to the equation $\sum \vec F=m\vec a$, where $\vec a$ is the translational acceleration of the center of mass. This applies whether or not the reaction force is in line with the center of mass.
In the cases where the reaction force is not in line with the CM, It will also add to the Torque equation $\sum \vec \tau =\sum \vec F \times \vec r= m\vec \alpha$.
Reaction force acts at the point of contact but for convinience of calculation we usually take it on COM
Think about it
As Newton stated, a "reaction force" is the equal and opposite of the action force. So, if an action force from an object originates in its center when colliding with another object, the reaction force will be directed towards its center.
When gravity pulls on an object, the action force is Earth's center pulling on an object's center. Thus, the reaction force is the object's center pulling on the Earth's center.
A reaction force is basically an electromagnetic force. When we push on a rigid surface, what we are doing is that we are decreasing the distance between two surfaces. This results in increase of the unbalanced electromagnetic forces on the two surfaces since they are not perfectly neutral thus the reaction force acts on these points at the surfaces which are pushed very closely to each other (which appears to be in contact when seen by a naked eye) so if you are analyzing a whole surface then you can say that the reaction force acts on the point of contact between the surfaces.
