Does the bible speak of "prayer chains" as we know them today?
Certainly the Bible does not speak of “prayer chains” as we know them today. That is, it says nothing about adding your prayer request to an e-mail list or web site and having people all over the world pray for that item. However, it does have instances of someone writing a letter and asking others to pray for them. Usually, though, the people asked to pray knew the person about whom they were praying.
Now I beseech you, brethren, for the Lord Jesus Christ's sake, and for the love of the Spirit, that ye strive together with me in your prayers to God for me. (Romans 15:30)
Continue in prayer, and watch in the same with thanksgiving; Withal praying also for us, that God would open unto us a door of utterance, to speak the mystery of Christ, for which I am also in bonds. (Colossians 4:2-3)
Brethren, pray for us. (1 Thessalonians 5:25)
There are others, but this should suffice to show that Paul approved of asking others in different cities to pray for him. It may not be quite the same as a modern prayer chain, but the idea is similar.