# Does group action on manifold induce subbundle of tangent bundle?

A Lie group $$G$$ acts smoothly, properly and freely on a smooth manifold $$M$$. We define $$S=\cup_{p\in M}T_p(G\cdot p)\subset TM$$. Is $$S$$ a sub vector bundle of $$TM\to M$$?

Yes. The projection onto the quotient $$p\colon M\to M/G$$ induces a morphism of vector bundles $$TM\to p^*T(M/G)$$ and the bundle you describe is the kernel of this map.