Is there a place for worriers in the church? Or is it a problem that some people perpetually worry?
Yes, there's a problem with that; and yes, there's a place for them, because there is a place for sinners in the church. A person who is content with being a chronic worrier should ask whether or not they are a Christian. But a person who has to battle worry every day--either because of the way they are wired or because of how they were raised--if they are willing to fight their worry then they are in the same category with those who fight against lust, pride, or bitterness everyday. We all fight something everyday. If we're not fighting something then we are being taken advantage of by our own sin.
I want to be careful to tell you that you must not treat anxiety lightly. Jesus said over and over again to not be anxious, either about what you should eat or drink or how you should speak about him. He promises to be with us, take care of us, and provide for our needs. He is always fighting fear and anxiety. Therefore, he clearly wants us to overcome it.
Likewise the apostle Paul says to not be anxious about anything, "but in everything by prayer and supplication with thanksgiving let your requests be made known to God. And the peace of God, which surpasses all understanding, will guard your hearts and your minds in Christ Jesus" (Philippians 4:6-7).
So the Bible clearly wants us to fight to overcome anxiety, but it knows that we have more or less success from day to day. Some people clearly have more of a struggle with it than others. And there is a place for those strugglers in the Kingdom. But don't be content with it. Grow in courage and grow in boldness.