The words translated “sick” in this passage are just as easily translated “weak.” In 1 Corinthians 8:9 it is translated “weak” and to translate it “sick” in that passage wreaks havoc to the argument. You can go back into that passage and try to make “sick” work, but it won’t. Considering the context in James, I think “weak” is a spiritual weakness, not a physical weakness.
It think it best to read this next to the text in your Bible. I think the flow of the encouragement James offers makes mores sense, but it must be recognized that every translation says “sick” here. I am not pushing so much to stop applying this to sick believers, but to encourage the church to apply it to those weak in faith or conscience.
James 5:13-16 Is any one of you in trouble? He should pray. Is anyone happy? Let him sing songs of praise. Is any one of you weak? He should call the elders of the church to pray over him and anoint him with oil in the name of he Lord. And the prayer offered in faith will make the weak person strong (preserve the weak one); the Lord will restore him. If he has sinned, he will be forgiven. Therefore confess your sins to each other and pray for each other so that you may be corrected. The prayer of a righteous man is powerful and effective.