How is the Euler-Lagrange equation derived without local coordinates?

The Euler-Lagrange equation states that the time flow is given by a vector field such that the vector field contracted with the symplectic form gives dL, where L is the Lagrangian function on the tangent bundle.

I don’t think your characterization of the vector field is quite correct; I would recommend Cartan (1922, §§184-185), Godbillon (1969, Proposition XI.3.7), or Souriau (1970, Theorem 7.29). Or this answer conveniently linked in this page’s “related” list $\to$.
• @Tyson 1) I posted “that” because as I said, it corrects your erroneous characterization of the vector field. Its contraction with the 2-form is not $dL$ but $-dH$ where $H$ is the Hamiltonian. 2) What matters in differential geometry is that results be coordinate-free. Better come to terms with the fact that not all proofs can or need to be. (E.g.: existence of the exterior derivative.) 3) That said, I think Souriau’s formula (7.28) (= 6th displayed in Cartan’s §184) gives about as close to an intrinsic argument as you’ll get. Call them “completely wrong” at your own risk :-) Dec 13 '17 at 12:17
Usually one thinks of the symplectic structure $\omega=d\lambda$ as living on the cotangent bundle $T^*N$ of a manifold $N$, where it is the exterior derivative of the Lioville 1-form $\lambda$. Hence, contracting it with a vectorfield will require that the vectorfield lives on $T^*N$ and the result will be a 1-form on $T^*N$. Since your Lagrangian $L$ lives on $TN$, $dL$ is a 1-form on $TN$ so something needs to be corrected.
The way one usually looks at this is the following: If the Lagrangian $L$ on $TN$ is Tonelli then it gives rise to a Legendre transformation $$l:TN \to T^*N,$$ which is a diffeomorphism. This allows one to transfer the dynamics of $L$ to the cotangent bundle. The way this is done is by defining a Hamiltonian function $H:T^*N \to \mathbb{R}$ via $$H(l(q,v))=\partial_vL(q,v)\cdot v-L(q,v),$$
where $\partial_vL(q,v)$ denotes the fiber derivative of $L$ and $(q,v)$ are coordinates on $TL$, $q$ being the base-coordinate and $v$ the fiber-coordinate. The Hamiltonian flow of $H$ is then defined exactly as you say: Namely it is the flow generated by the vectorfield $X_H$ whose contraction with $\omega$ equals $dH$. The vectorfield $X_L$ on $TN$ which generates the dynamics dictated by $L$ is now obtained by $$dl \cdot X_L=X_H.$$ Of course one can also derive the fomula for $X_L$ using the principle of least action.