The book of Hestenes does not give a definition of vector manifold. It says that a vector manifold is a set of vectors in a Clifford algebra, with some additional properties. It defines those properties in terms of a notion of interior, and a notion of boundary, neither of which are defined. So it relies on intuitive notions of interior and boundary. It defines tangent space at a point (of any set of vectors), as the set of velocities of curves through those vectors. Presumably the curves are required to be smooth enough to have velocity vectors. It states that at interior points, the tangent space is a vector space. But that can't be enough to define a manifold in the usual sense, because it doesn't say what an interior point is. A set of vectors much worse than a submanifold can have points at which all velocities of differential curves in the set are zero. So the property of having a vector space as tangent space does not decide for us how to define the notion of interior point. In the end, the authors are relying on prior experience with manifolds, especially surfaces, as is often the case for authors working close to classical physics, engineering or statistics.

So to answer the questions: there is no coordinate free definition of manifold, because every definition we currently have relies on some charts, or on being a submanifold of some other previously defined manifold, ending us up with Euclidean space. There is no way to fully justify Hestenes's definition, because he doesn't really have one, but we can say that Nash's embedding theorem proves the existence of isometric embeddings of Riemannian manifolds, and subsequent authors have generalized to pseudo-Riemannian manifolds.