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. See Bottle Skip and Cabana Deed Skip.
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 (which would crash at that point due to animation sets). This means you must perform the skip twice in the run.
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 skip the delivery bag with a zombie hover, 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).
Using chest storage and the grappling hook, you can climb the walls up to Medli and skipping having to go inside before the Earth Temple. Using the 200 rupee chest that you can get with a bow shot, this is a really fast way to skip the delivery bag after obtaining the grappling hook.
It is also possible to skip the delivery bag with chest storage and without the grappling hook. First, get chest storage. You can use a hyoi pear to get the 200 rupee chest, but if you are running low% you may want to get another chest. With chest storage, you can activate Quill's cutscene early to get higher on the island, and then you can climb the wall on the left side of the loading zone giving you the delivery bag. Once on the higher floor, enter the loading zone to skip the delivery bag.