From another thread I wrote this in, I decided to make a separate thread with what I wrote, just so people will have a general go to place to learn how to set up Brawl mods for use with Dolphin.
You need the following, alongside your folders set up.
Virtual SD Card Maker: https://sites.google.com/site/anon4453/tools/easy-dolphin-sdcard-maker
Gecko OS: http://wiibrew.org/wiki/Gecko_OS
A later version of Dolphin Emulator: https://dolphin-emu.org/download/
You used to need to download a specific IOS with an NUS Downloader, but I don't think you need to do that anymore with later versions of Dolphin.
This was the version I previously used: https://www.smashladder.com/guides/view/264m/dolphin-ladder-build-4-0-7840Edit: Please note that this specific version is quite outdated, but is probably still used for more optimal Netplay experiences.
1.) Create a Virtual SD Card using the Virtual SD Card Maker program. You can go as low as 64 MB or as high as 2 GB, though 2 GB is highly recommended (I believe you can use a cmd function to make the file over 2 GB, but this is just for the basics). Make sure the file produced is named sd.raw and place the file under "C:\Users\*Name*\Documents\Dolphin Emulator\Wii". If there's already an sd.raw file in there, delete it before inserting your newly created file.
2.) Open WinImage and open your sd.raw with it. At this point, you should have all of your files, codes, and Gecko OS inside their respective folders as if you were going to use a Wii. You would also place everything into a dummy folder (Just create a new folder and put everything in it). In WinImage, go to Image > Inject a Folder and search for your dummy folder before injecting it. Save and close afterward. You'll also need to place the boot.elf file from Gecko OS into a folder of your choice.
3.) Now in Dolphin, right click on your Brawl entry before clicking on "Set as Default IS0." After that, open up Config, go to Wii, and make sure Insert SD is enabled.
4.) To boot modded Brawl though Dolphin, you can do one of two things. Though before starting the game, make sure you configure Gecko's Hook Type to be AXNextFrame.
4a.) Click Open before searching for your saved boot.elf file.
4b.) Browse for the folder that contains your saved boot.elf like if you're going to add a new IS0. This entry will stay in your Dolphin window for quicker access.
If there's anything else that needs to be added, let me know, and I'll update with misc information/fixes.