The word chapel can be found in one place in the King James Version (KJV) of the Bible. It is a rather obscure verse in Amos, chapter 7, verse 13. It says,
“But prophesy not again any more at Bethel: for it is the king’s chapel, and it is the king’s court.”
I could not find it in any of the other six common translations. In the same verse, the word sanctuary is used instead, a word which is also used in many other places in the Bible, and will be studied at another time. (See blueletterbible.org.)
The origins of the English word chapel are from the story of a Roman soldier who is said to have shared part of his cape with a beggar. He supposedly had a dream after that generous action, in which he saw Jesus wearing that shared part of the cape. As a result of telling of the dream, the soldier’s portion of the cape was set aside in a special room, where the capella, Latin for “little cape”, could be viewed in reverence. From there, the word evolved to mean a “smallish, more private room where more personal meditations could take place.” (See etymonline.com and Webster’s New World College Dictionary, Indexed Fourth Edition)
As such, a chapel is generally considered to be an extension of more officious “church” buildings. The room is sometimes attached to these buildings, but other times it is set up in places where people might want a religious room, but cannot easily get to a more established building. Thus, chapels are often in hospitals, in the battle field, or in schools.
Some people use the word chapel synonymously with sanctuary, possibly trying to make use of the more personal implications of the word chapel. Or maybe just looking for names of buildings that haven’t already been taken. Either way, it does not have any Biblical roots, the soonest usage recorded being the Late Latin period. Of course, that doesn’t mean we aren’t allowed to use it, but if anyone is looking for Biblical precedent for chapels in early Christianity, it doesn’t seem to be there.