As far as I'm aware there is only the option to copy and paste a slot but no file that it can be stored in.
You can however, put your exanimation as level animation, then export the level itself to a file. This file will have the entire level and it's settings, but also the exanimation you care about in it.
The user would have to open this file, go to the exanimation window and press Copy Slot or Copy All Slots, then go to their ROM and press Paste Slot/Paste All Slots.
This is usually far less work then actually having the user set everything up themselves. Most submission including exanimation I've seen or done use this method and I'm not aware of any LM updates that make this easier.