You can upgrade the Bomb Bag's capacity by performing a couple of tasks.

Upgrade	How to unlock
Big Bomb Bag (30 Bombs)	As a Hylian, stop Sakon from mugging the Bomb Shop Lady on the First Night. It'll be for sale for 90 Rupees at the Bomb Shop.
Big Bomb Bag (30 Bombs) (Alternative)	Alternatively, if Sakon succeeds in stealing the Big Bomb Bag, it goes on sale at the Curiosity Shop for 100 Rupees on Night 3.
Biggest Bomb Bag (40 Bombs)	As a Goron, talk to the Business Scrub in Goron Village. Trade the Big Bomb Bag and 200 Rupees for the upgrade.