The question is answered by Proposition 4 in this paper by Auroux. The Proposition says that given a family of symplectic submanifolds $\{X_t:t \in [0,1]\}$ in a symplectic manifold $M$, there is a family of symplectomorphisms $\Phi_t:M \to M$ such that $\Phi_0=\operatorname{Id}_M$ and $\Phi_t(X_0)=X_t$. In the question I had asked for a Hamiltonian diffeomorphism. A sufficient condition to ensure this is if $M$ is simply connected. In that case the family $\Phi_t$ is generated by a Hamiltonian vector field.