For vfat file systems just use mcopy. Then you can add without mounting. But that’s if you want to do file copies into one. For linux I’ve used the fuseext2 to mount in user space and that works.
To a first approximation, no; the usual practice is to mount any raw images or disks on the host system and then push them in with docker run -v options. Like the network, the filesystem setup is something that the Docker runtime takes care of and the process never thinks about.
It sounds like this task might not be a good one for Docker, and a VM or host-native process might be better.
The only way to get into those images is to mount the image in user space. xorriso also is good for getting info out of iso images. But I’ve had to use mcopy for vfat, fuseext2 for ext2 filesystems, and xorriso to extract ISO images. Mostly because my application is supposed to do this. I’ve had to work around not being able to mount images. But those have worked good enough for us. mcopy in particular is really small so makes the container not as big.