Theorem: Let $Q_i\to (X_i,w_i)$ for $i=1,2$. be symplectic submanifolds in a symplectic manifold. Suppose there is a symplectic identification of their symplectic normal bundles $F:\nu Q_1 \to \nu Q_2$ which covers a symplectic isomorphism $Q_1 \to Q_2$. Then $Q_1$ has a neighbourhood symplectomorphic to a neighbourhood of $Q_2$.
The idea of the proof is as follows:
Fix metrics on $Q_i$. Use exponential maps $g_i$ to identify symplectic normal bundle with neighbourhood $N_i$ of $Q_i$. And then consider $G:=g_2 \circ F \circ g_1^{-1}$. After showing that $G$ pulls back neighbourhood of $Q_2$ to $Q_1$ and preserves symplectic structures on $T_qX_i$ for all $q\in Q$, the result follows from a technical lemma proven before.
I don't quite understand couple of points. What is a map $F$ ? Namely, what does it mean for $F$ to be symplectomoprhism? Do we think about $\nu Q$ as a manifold and we endow it with a symplectic structure after identifying with a neighbourhood of $Q$ in $X$ ? Or do we think about $\nu Q$ as a symplectic bundle over $Q$ as a subbundle of $TX|_Q$ ?
It seems to me that it is the second case. Is $G$ restricted to $Q_1$ a symplectomorphism for the following reason? $T_q N_1$ decomposes as $T_q Q_1$ and the normal complement. The isomorphism $T_q Q_1 \to T_{G(q)}Q_2$ follows because $F$ covers a symplectomorphism and the isomorphism on the normal complement follows because of the requirement for $F$ to preserve symplectic structure.
P.S. One of the steps wept under the rug is introducing an almost complex structures on $X_i$ such that $Q_i$ are pseudo-holomophic. This allows us to identify symplectic and orthogonal normal bundles.