Since you'll have MIDI anyway, you might want to use it to control all of the pneumatic valves. i.e. Each animatronic character would be similar to a MIDI instrument but responding with movement rather than by generating sound.
Normally, something like that would more-likely use DMX (a theater lighting & effects standard) but MIDI and DMX are somewhat "similar".
I assume you'll need to have a separate MP4 and control files and start them at the same time to keep them synchronized. I'm also assuming that the MP4 microSD or plugs-into the Sprite player, and the Arduino would have its own memory for the animatronic sequence.
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I actually worked at the Chuck E. Cheese factory where those things were made. The most fun job I ever had!!! I remember the pneumatics but I don't know exactly how the control worked. I believe it was "analog" tape. Probably "stereo" with the audio on one channel and the control signals on the other channel. There are (were) ways of putting digital on analog tape, similar to how the old modems worked over the phone line. Around the same time I had a Commodore computer and a cassette player/recorder was standard for storage, but I had a floppy drive.