Here's a little trickery that I did and was able to add my custom audio as a Yahoo notification sound without adding any third party ringtone apps. This does require root access, though.
My logic was (could be flawed, but it worked!) that the Yahoo Mail app was pointing to system audio/ringtone files and didn't have it's own file collection, hence not being able to just dump some new audio files into a folder and have them show up as options.
So I just overwrote one of the system notification sound files with a custom sound of my own that had the same filename.
1) From within the Yahoo Mail app, identify one of the notification sounds that you're sure you'll never want to use. I chose the "Caffeinated Rattlesnake"
2) Locate this audio file in your system folder. On my phone (Droid Razr Maxx, Android 4.1) it was in /system/media/audio/notifications and was named "CaffeineSnake.ogg"
3) Now use an audio program (I used Audacity) to create an audio file of the new custom sound that you want to use. Save/export that file as an "Ogg Vorbis File" and name it "CaffeineSnake.ogg"
4) Get the new "CaffeineSnake.ogg" file onto your phone, then move it to the /system/media/audio/notifications folder and overwrite the existing "CaffeineSnake.ogg." This step requires root access to overwrite the system file... I use the app 'Root Explorer' by Speed Software to access/copy/paste/overwrite system files and folders.
5) Now go into Yahoo Mail and go to the notifications section and choose the sound "Caffeinated Rattlesnake."
Voila! You'll now hear your custom sound when a Yahoo Mail notification comes through.
Hope this helps!