Welcome to AskUbuntu. Have you already seen previous questions like this: Is there a way to install Ubuntu on usb pendrive as normal instalation (not live cd)? of which the provided answers clearly states that it is possible to run Ubuntu from both USB and/or SD Cards.
I once wrote this as an answer to that question:
Based on what @zuberuber says you can even install it on SD Card. With
the proper adaptor or natively on sd card embedded laptops. This is
how I am writting this: Installed in a 16GB SDCard with 10.5GB Free.
Anyway I suggest you to disconnect all your hard disk drives in your
computer, specially those which are having a OS installed (Windows,
Linux, etc.) as you may finish having a multi-boot USB/SD Memory with
GRUB and in certain cases the bootloader can be written in a different
device, other than the USB Memory/SD Card.
Additional answers to this and other similar usb/sdcard related questions can be very illustrative.
BTW: The boot from USB/SD Card can slightly be slow compared to a hard disk drive install but after the boot you won't experience any speed issues at all.