This page has a helpful summary of the history--it seems he initially accepted the Aristotelian idea that objects could only continue to move if some "force" inside them was moving them (keep in mind this is before his technical definition of 'force'), and it took him a while to switch to the idea that bodies naturally tend to keep moving unless acted on by a force, i.e. inertia (it seems he got this idea from Galileo and Descartes). After this he developed the concept that "force" must be acting whenever there is a change in motion, i.e. an acceleration. From there, the article suggests he got the idea for the third law from various mechanical experiments in which it could be observed that the total momentum always remained constant (and if you define force as mass*acceleration, conservation of momentum implies that forces must always be equal and opposite):
Continuing his investigation of impact, he analyzed a collision
between two bodies of unequal mass in the center of gravity frame of
reference. He stated that they had “equal motions” in this frame, both
before and after the collision. This could only mean mass×speed, or
momentum (equal and opposite, of course)—the Third Law. (He realized
and stated that during such a collision, the center of mass itself
would move at a steady speed.)
A Third Law Experiment with Pendulums
In fact, there is a Third Law experiment in the Principia, in the
second Scholium, right after the Laws of Motion and their Corollaries.
He collided together two pendulums (about ten feet long) with
different masses, to establish that the impacts (i.e. forces) felt by
them were equal and opposite, as measured by how far they rebounded.
He went to considerable trouble to account properly for air resistance
and imperfect elasticity.