Although the Delivery Bag is used to carry a few items, particularly the Father's Letter for Komali, and the Cabana Deed, the need to obtain these items can be avoided by using a couple of useful techniques to skip them.
Note: In order to be able to enter Dragon Roost Cavern after performing Delivery Bag Skip, you must either have Bombs Early, a Bottle early, or the Deku Leaf early. Also, in order to fully skip the Delivery Bag, you must also avoid the cutscene trigger on the way to Medli before the Earth Temple. The easiest way to do this is to get chest storage on the 200 Rupee chest at Dragon Roost and then use the Grappling Hook and Leaf in order to climb up to Medli on top.
The cutscene in which you would normally obtain the Delivery Bag is notoriously long, tedious, and just another good reason why skipping the item is a useful timesaver. In order avoid this cutscene's trigger, you will need to reach the upper level of the main lobby from the outside. To do this, lower your health to 1/4 heart and make sure to have a Red Ting handy (using the bomb flowers to lower your health would be preferred, as it saves explosives). Now go over to the shrine where you learned Wind's Requiem. Stand between the two stones and turn around. From here, zombie hover to the upper platform and heal yourself with a red ting once above it. This also skips the Quill cutscene you would normally encounter. From the upper ledge of the main lobby, roll and grab the ledge and then head over to the Dragon Roost Cavern entrance. From here, either leaf across the gap (if you have Deku Leaf early), pour water on the bomb flower (if you have a Bottle early), or Zombie Hover into Dragon Roost Cavern (if you have Bombs Early).