A surjective submersion can be characterised in the category of smooth manifolds as being an element in the largest pullback-stable class of regular epis. Also, submersions have local sections through every point in their codomain. Or they look locally like a projection out of a cartesian product with fibre a vector space. This latter is in fact how submersions are defined for infinite-dimensional (well, Fréchet) manifolds. The first definition works fine for smooth spaces, but proofs may not generalise if they need some of the other properties. The second one works if you are considering concrete smooth spaces, but is meaningless otherwise. The third definition implies the second, but restricts the sort of fibres.
Edit: Another property of surjective submersions is that they are the largest class of morphisms which admit sections over open covers (which we can assume are made up of charts) of which all pullbacks exist. This means that surjective submersions are in some sense a 'saturation' of the singleton pretopology consisting of covers of the form $\coprod U_i \to X$.