It depends on how you define manifolds and what properties you are allowed to assume.
My favorite way is to use things like the constant rank theorem which allows to say that a manifold M embedded in euclidean space $R^n$ is locally the set of zeroes of some functions $f_1,\ldots,f_n$ and that the tangent space at a point $p$ is given by the subspace orthogonal to the derivatives of the $f_j$. (The set of vectors orthogonal to some set is always vector space).
On the other hand, combining the approaches given in the other two answers, if you have a parameterisation of your manifold $\varphi: R^d \to R^n$ around a point $p$ then the axioms of a vector space can easily be checked. First you have to define scalar multiplication and addition (and then associativity and distribution law follow from properties of derivatives).
Given a smooth curve $c: I \to M$ (where I is an open interval around zero), with $c(0)=p$ and c'(0) = v, you can define c'(t) = c(\lambda t), which defines scalar multiplication. (In fact, dc' = \lambda dc, by linearity of the differential.)
Given two curves $c_1,c_2: I \to M$, consider $\psi \circ c_1, \psi \circ c_2$, where $\psi = \varphi^{-1}$ is the inverse of the parameterisation. Finally consider the curve $c = \varphi \circ (\psi \circ c_1 + \psi \circ c_2)$, this defines addition. (In fact $dc = d\varphi d\psi dc_1 + d\varphi d\psi dc_2 = dc_1 + dc_2 ,$ by additivity of the differential and the fact that $d\psi = d (\varphi^{-1}) = (d\varphi)^{-1}$.)