I couldn't find a USB floppy device implementation, so I wrote one.
This was written for the Atmel/Microchip SAMD11 and SAMD21 USB microcontrollers.
The implementation follows the USB Mass Storage Class CBI Transport Specification and USB Mass Storage Class UFI Command Specification.
The source code provides just enough emulation of the file system to fool the PC host, but you could add your own non-volatile storage implementation by replacing virtual.c (You'll also have to make your own 'chu-chunk, chu-chunk' sounds as you copy files if you want to hear that too.)
The USB stack source code owes its origins to vcp.
One approach is to use Rowley Crossworks for ARM to compile this code. It is not free software, but has been my favorite go-to ARM development tool for a decade and counting.
OR
Use the Makefile in the make subdirectory. With this approach, the code can be built using only open-source software. In Ubuntu-derived distributions, this is likely achieved with as little as:
sudo apt-get install gcc-arm-none-eabi libnewlib-arm-none-eabi build-essential
Testing was done with Linux, Windows 7, and Windows 10.
All versions of Windows require a .inf file for VID:PID values other the ones natively supported by the OS. "Wait", I hear you say, "don't you know Windows support all Mass Storage devices regardless of VID:PID?" The wisdom of the Internet is wrong. Redmond may have given up trying to monetize MSC Subclass 06h, but their money-making machine apparatus is alive and kicking.