Lots of Teensy-powered things have microSD; for instance, Radio Music from Music Thing Modular:
The latest, not-so-teensy Teensy have a built-in SD card reader and headers for a second one:
YES! Though, you can actually buy a bootloader chip from PJRC so that you can put just what you need on your board, avoid having to piggyback the Teensy, and still use the bootloader ecosystem. Just don’t sneeze while you are working with the chips:

https://www.pjrc.com/store/ic_mkl02.html
Finally, Oliver at Mutable has MIT Licensed his Audio Bootloader that allows you to build a product that a user simply has to play an audio file into your device and the firmware can update.
Cool stuff!!