There are several different ways to obtain Bombchus before completing Dodongo's Cavern, the point at which the Bombchu Bowling Alley and Bombchu Shop open.
The two most common ways to obtain Bombchus early are to get them from the Bottom of the Well (Well Chus), or to get them from the Spirit Trial in Ganon's Castle after performing Ganondoor.
In Master Quest only, you can also obtain Bombchus from Jabu Jabu's Belly, provided that you have a slingshot.
Another option is to obtain Bombchus from somewhere within Gerudo Desert. There are Bombchu chests inside both the Spirit Temple (child side) and the Gerudo Training Grounds. Both areas can be entered early using various tricks to get past different barriers within the desert (see Spirit Temple Early for more info). In the original game, getting Bombchus from the Gerudo Training Grounds is not practical as a child, and if you don't have the Slingshot or Boomerang to hit a crystal switch for the Spirit Temple chest, you will have to hover off a Lizalfos to reach it. However, In Master Quest both chests are quite easy to reach.
The last place Bombchus can be obtained early in the desert are from the Carpet Salesman in the Haunted Wasteland. Since Chus cost 200 rupees per 10, you will need to collect 10 Gold Skulltula tokens to unlock the Adult's Wallet in order to purchase them.
Finally, the reward for collecting 40 Gold Skulltula tokens is 10 Bombchus. This is the most absurd way to obtain Bombchus early, moreso if you do not resort to duplicating Gold Skulltula tokens.
Once in, one needs a way of getting to the crawl space near the entrance under the water. Zelda's Lullaby can be played at the other side of the well to lower the water, but in a speedrun, one would typically want to get bombchus before heading to Hyrule Castle. It is possible to use Walking While Talking to perform an Actor Glitch to unload the water, thus bypassing the requirement of Zelda's Lullaby.
First, one needs a key, which can be found by heading around the corner to the right of the entrance from a chest behind a fake wall. Proceed up the right hand pathway to find a crawl space and enter it. When exiting the other side, one must open the door before the "exiting crawlspace cutscene" ends. The analogue stick position for this angle is very slightly down from down-left. As soon as control is regained, hold this position and roll to the door and press A to open it. If you time it and angle it correctly, you will hear the door opening but you will remain in the same room. If you are too late, you will simply go through the door.
In performing this trick, the actors in the next room are loaded while the actors in the main room are unloaded. Included in these unloaded actors is the water near the entrance, allowing one to access the previously unreachable crawl space. Crawl through here to reach Dead Hand. By defeating Dead Hand and exiting his room, the correct actors are loaded again and one can Vine Clip using the Skulltula.
Once Dead Hand has been defeated and you have reloaded the Skulltula, the final step to getting to the bombchu chest is to clip out of bounds and swim through the invisible water. There are two methods of clipping, one with Deku Nuts and one without. The method without is the most common as it can easily be pause buffered and lands you in the high water, whilst the Deku Nut method is not normally buffered, but lands you in the low water.
Once clipped through, you land in invisible water - either "high" or "low" water. High water is far superior because it is possible to swim directly to the bombchus without having to walk around out of bounds. If one lands in low water, one will fall to the ground and will have to navigate to a specific spot where it is then possible to float up to the bombchus. This is harder and slower, so is not recommended.
Using Deku Nuts, which can be found in a chest in the central area of the main room, one can stun the skulltula and drop and regrab the vines just as stun timer on the skulltula wears off.
If you fall out of the water whilst swimming to the bombchus, it is still possible to float up in a column of invisible water just below the room containing the bombchu chest. Head over to the strange looking floor texture and locate the sharp point. Just at the tip of this point is a column of water which can be used to float up to the bombchus. Backflip into it, float up for a couple of seconds, then swim back-right whilst holding Z. If done correctly, Link will appear to jilt a little bit, which means you have clipped through the floor. From here, you can float up a little bit more before swimming forwards and jumpslashing out of the water and in to the room, which by now should be visible on the screen.
Often you will get pushed out of the water when you try to backflip in - just try backflipping in again. Sometimes Link will not clip through the floor, in which case just try adjusting the time at which you start swimming down-right and try again. If, once you clip through the floor, you wait too long, then you will float all the way back up to the main room and will have to start again from the Actor Glitch. If you walk too far on to the texture, it can damage you, so avoid doing that.