EDITOR'S NOTE: This question is NOT about the Covid-19 vaccine. Pastor Paul is responding to a general question about vaccines, medicines and doctors and how such things fit into a life of faith.
What might be an act of unbelief for one person isn't necessarily an act of unbelief for another. It all comes down to the heart.
For example, one man may submit to a surgical procedure but place himself completely in the hands of God for the outcome, while another man has the same procedure and places his complete trust in doctors and medicine for the desired outcome. It depends on the individual and where their ultimate trust and confidence lies.
Obviously the Bible doesn't specifically address vaccines or even say much about doctors for that matter. But it does talk a lot about trusting God. I suppose all those passages about trusting God with all of our heart are why some claim that going to the doctor and taking medicines is not walking by faith. But those very same people walk into grocery stores and clothing stores every day and think nothing of it even though Jesus promised that God would feed and clothe us. (Matthew 6:25-33) Wouldn't that also be considered a faithless act in light of God's promise?
I don't believe using doctors or medicines or vaccines are what define our faith. Real faith is a matter of the heart. One man may take a vaccine and remain fully convinced that God is in charge of his life and health, while another man expresses this same faith by rejecting the vaccine. Each one should be fully convinced in their own heart and not cast judgment upon the other.