Is this anything like the type that operates with a desktop steam engine? These might be available from suppliers of scientific toys (example, Edmund Scientific).
An aquarium air pump might be what you want. There are vibrating (inexpensive) types, and piston-cylinder (higher priced) types.
Old-fashioned bug sprayers are piston pump action (although the outlet is not a valve but a spray opening).
You can make your own pump, by placing flaps across holes in the end of a cylinder. Arrange the flaps to lift and seal in such a way that fluid travels in the desired direction. To make the piston requires ingenuity. This was often a disk which had a leather washer attached to act as a seal.
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Now I see messages were posted while I composed mine. I was off track, I find.