As a geometry object, the definition of a foliation doesn't depend on the choice of the charts. The foliation defines a collection of n-q dimensional sub-manifolds. In each chart (x,y), a plaque is defined as y=c where c is constant, though in another chart (x1,y1) the same plaque is represented by the same type of equation y1=c1, with a (possible) different constant c1. So the plaque( a part of a submanifold) is defined independent of charts. Then the tangent space of this plaque is well-defined. Piece together all those tangent spaces we get the sub-bundle.
To show it's involutive you need the following equation:
Let and notice that , we get , which shows [X,Y] belongs to the sub-space.