A charge will experience a force when it travels in a magnetic field. By Newton 3rd law, what are the action and reaction pair?

The action force that experienced by the moving charge is given by the magnetic field. Hence, the reaction force acts on the field, which is experienced by the object that produces the magnetic field.

Just think of a situation that is most familiar. Suppose there are two electromagnets (or solenoids) with current in each of them flowing in opposite directions. If the two electromagnets are placed end to end facing each other, they will experience a repulsive force due to interaction of magnetic field produced by each of them. The pair of repulsive forces that act on the electromagnets constitute an action-reaction pair.

Now, suppose there remains one electromagnet, and a charge is moving across its magnetic field. The moving charge generates a magnetic field. This magnetic field will interact with the field of the electromagnetic. Similar to the above example, the interaction of the two fields (fields produced by moving charge and electromagnet) produces an action-reaction pair, one acts on the charge and the other on the electromagnet.

one on the electron, another on the object which providing the magnetic flied.