How much influence does he have?

[tackattack]

To answer the Op's underlying question. Yes you.. as an atheist could be used to fulfill God's will in a believers life. Typically that's covered in Ephesians 1:11-12 or more commonly Romans 8:28

How is that covered? Explain it to me, please. I could go with my own interpretation of these scriptures but it would be biased towards seeing God as a bad faith actor. So,  tell me what these mean to you and how it relates because I'd rather not devolve into assuming what you're saying.

The question isn't "could I" it is "how": how does God use people to answer other people's prayers? Because I am concerned about the agency of individuals and why God can exert power on people in some circumstances (not every prayer is answered by a person showing up at the right time; sometimes nobody shows up).

I personally believe God is in control of everything. He permits things according to His infinite wisdom and grace. God works through confirmation by other believers, God works in circumstance, God works by the urging of the Holy Spirit, God works by direct revelation and prayer, and all of those ways align with the Word of God and work for God's glory or it's not from God. 

I don't believe God sends little nanites into your brain to forcibly change your mind, but I do believe an all knowing God, knows that your decision to do X will affect those around you Y way and allow for opportunities for Z within your sphere of influence and that produces actionable cause and effect in people's lives. God can use your selfish or selfless actions for His glory regardless of your intent. Just like the story of Joseph. His brothers intended him harm, but it worked out for the glory of God. God is sovereign over everything, and while it may be predestined from His perspective it seems free to us, not that He controls our whims, but He uses both our good and evil for His purposes.