Question: How does Objectivism reconcile the idea of free will with its strict physicalist doctrine?
Answer: Objectivism does not have a “strict physicalist doctrine.” It merely holds that whatever exists exists, has identity, and has causal powers. So if this is “physicalism” or “materialism,” it is a richer example of those doctrines than is normally thought, since disembodied minds (if they existed) would be perfectly compatible with the Objectivist metaphysics. What Objectivism denies is supernaturalism: the idea that there is somehow a reality or part of reality that is not really connected in any comprehensible way to our universe or that is excepted from the (minimal) constraints of identity and causality.
Perhaps you mean by your question: how does Objectivism reconcile free will with causality? The answer to that question is that Objectivism has an Aristotelian, entity-causation view of causality. Causality is the law of identity applied to actions, and actions are actions of entities. It is entities, i.e., things, that act. A ball rolls, an atom decays, etc.
In the case of human beings, our causal mode involves free choice over a range of actions. We each choose to think and act. This is a natural property of human beings, arising out of the structure of our brains, it appears. Certainly, damage to the brain can eliminate this faculty (as evidence, consider not just the death that follows from most brain trauma but the sad phenomenon of a person rendered a “vegetable” by an injury to the brain).
Free will is metaphysically no more mysterious than any natural phenomenon. Hopefully in time psychology and neuroscience will succeed in understanding better how the brain gives rise to volitional consciousness and how free will functions. But that we have it is self-evident to those who do.
Objectivism rejects the religious and post-religious dualism associated with Rene Descartes. This view regards causation as a billiard-ball cascade of events, and holds all the material realm to be necessitated or determined by previous events. The mind, and free will, in Descartes' view, must exist is some radically different and fundamentally disconnected spiritual realm. Objectivism sees no need for this on metaphysical grounds. In fact, there are many objections to it. How could the realms determined matter and free spirit relate? What other evidence is there for a mental realm, anyway? In fact, the Objectivist approach is much more sensible and cuts through Descartes' self-defeating conundrum.