In the book Tom Bombadil, if nothing else, rescues the hobbits from the Barrow-Wights & gives them knives from the barrow as swords. Frodo uses his to attack the Lord of the Nazgǔl at Weathertop. Merry more successfully uses his to help kill the Lord of the Nazgûl. No Bombadil, no swords.

There are other ways they can get swords, which they do in the films (Strider gives them to them). Another option is to simply have them find them on the corpses of previous victims in the barrow and get them themselves.