Yes, God certainly can use natural disasters to speak to us -- just as He can use other difficulties and tragedies to turn our hearts toward Him.
What can such events tell us? For one thing, they remind us of the brevity of life. We may be strong and successful, and assume life is always going to be that way -- but when disaster strikes, we realize this isn't true. We can lose everything in only a few seconds, and perhaps for the first time we are brought face to face with the reality of death -- and our need of God. Isaiah wrote, "Surely it was for my benefit that I suffered such anguish. In your love you kept me from the pit of destruction" (Isaiah 38:17).
Disasters also can remind us of our need to help others, and not just be concerned about ourselves and our problems. Not long after Hurricane Katrina devastated large parts of New Orleans, my son Franklin and I visited the affected area. I will never forget the way that disaster brought people from different backgrounds together, especially in the churches. The Bible says, "Carry each other's burdens, and in this way you will fulfill the law of Christ" (Galatians 6:2).
We don't necessarily know why God allows natural disasters to occur; sometimes Satan seems to have a hand in them. But the time to prepare for life's crises is now, not when they strike. Is your faith and trust in Christ, and are you seeking to live for Him every day, no matter what happens?