= Solution
Fix $p\in X$. A linear change of coordinates first identifies $\omega_p$ with the <standard symplectic form> $\omega_0$. After shrinking to a star-shaped neighborhood, every
$$
\omega_t=(1-t)\omega_0+t\omega
$$
is nondegenerate. The <Poincare lemma> gives $\omega-\omega_0=d\sigma$, with $\sigma(p)=0$. Define $X_t$ by $\iota_{X_t}\omega_t=-\sigma$ and let $\phi_t$ be its local flow. <Cartan's magic formula> gives
$$
\frac d{dt}\phi_t^*\omega_t
=\phi_t^*\bigl(d\sigma+d\iota_{X_t}\omega_t\bigr)=0.
$$
Thus $\phi_1^*\omega=\omega_0$, proving the <Darboux theorem (symplectic geometry)>.
For a smooth function $H$ on a closed symplectic manifold, its <Hamiltonian vector field> is defined by $\iota_{X_H}\omega=-dH$. The same formula gives $\mathcal L_{X_H}\omega=-d^2H=0$, so the <Hamiltonian flow preserves the symplectic form>. To move one point to another in a connected $X$, join them by a path, cover the path by finitely many <Darboux charts>, and in each chart use a cutoff linear Hamiltonian to perform a small translation. Composing these compactly supported Hamiltonian diffeomorphisms proves that \b[symplectomorphisms act transitively on each connected component].
In $\mathbb R^2$, every embedded curve is a <Lagrangian submanifold>. Let $L_1,L_2$ be circles enclosing different Euclidean areas. A plane symplectomorphism preserves area and carries the bounded complementary component of one circle to that of its image, so no symplectomorphism maps $L_1$ to $L_2$.
The same phenomenon exists in every $\mathbb R^{2n}$. With
$$
\lambda=\frac12\sum_{j=1}^n(x_jdy_j-y_jdx_j),
$$
take the product tori
$$
L_1=S^1(1)^n,
\qquad
L_2=S^1(\sqrt2)^n.
$$
The <Liouville class of a Lagrangian submanifold> has respective period vectors $\pi(1,\ldots,1)$ and $2\pi(1,\ldots,1)$ on these tori. Every symplectomorphism of $\mathbb R^{2n}$ preserves the Liouville class up to the induced integral change of basis on $H_1(T^n)$, because its pullback changes $\lambda$ only by an exact form. An integral automorphism sends a primitive vector to a primitive vector and therefore cannot send the first period vector to the second. Hence
$$
\boxed{L_1\text{ and }L_2\text{ are compact connected Lagrangians not related by any symplectomorphism}.}
$$
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