Yes, it is possible. You need to use an alternative method of booting games, called Loadiine:
https://gbatemp.net/threads/loadiine-gx2.413823/
The best way of doing this is to copy loadiine to your SD card and costumize your Haxchi config.txt file to boot loadiine when you press a certain button. For example, you could set up SDROOT:\haxchi\config.txt as such:
a=wiiu/apps/homebrew_launcher/homebrew_launcher.elf
b=
x=wiiu/apps/loadiineGx2/loadiine_gx2.elf
y=
default=wiiu/apps/mocha/mocha.elf
Now, whenever you keep the "x" button pressed when lauching haxchi, it will boot Loadiine GX2 (and you can press "a" to lauch HBL or just let it boot Mocha by default... or not, your choice). Keep in mind that running games from the SD card is... troublesome. Some games work fine, others suffer from bad load times. One prime example of this is The Wonderful 101, which can take up to 20 (!) minutes to load the initial menu from SD. If you install it through usb, it takes about 1 minute.
Loadiine compatbility list:
http://wiki.gbatemp.net/wiki/Loadiine_compatibility_list
My advice is, forget the SD card. Just get a cheap USB stick, format it to Wii U standards and use that. All games work perfectly from USB, its just like you bought them from the eShop.
