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 to 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).
There are a few setups for this hover:
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. However the grappling hook is past the delivery bag, thus skipping the delivery bag cutscene with chest storage involves not using that grapple spot. There are a few ways to do so.
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. Climb up the boulders and pick up a bomb flower to bomb boost up the ledge before the sidle gap. Now you can climb the wall all the way to the upper loading zone.
The bomb boost can also be done swordless with a backflip bomb boost.
Another, slower, method is to 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.
It is possible to heal from a zombie hover without the Tingle Tuner using a trick known as Fake Death. To do so, first get storage and store KoRL's text. Then die and zombie hover to the upper ledge of DRI. Once you are right above ground, pull a bomb, land and die. The death cutscene will start and KoRL's textbox will pop up on screen. Around the end of the death cutscene, close the textbox to get pushed off by the previously frozen-in-place bomb. Zombie Hover and wait for the "save the game" prompt to appear. Select don't save and then continue to get healed, skipping the delivery bag.
By charging a superswim, then going up an island (for exemple Outset) and activating a Tingle Balloon, you can go backwards at very high speed through the air. This can be used in a TAS to skip the delivery bag.