Here are some general comments. We can let $\Sigma$ be any closed $k$-manifold.

Fix a basepoint embedding $\Sigma \to X$.
Let $N$ be a compact regular neighborhood of $\Sigma$ in $X$. Then restriction defines a fiber sequence
$$
E(N,X) \to E(\Sigma,X) \to \text F(\Sigma, BO(n-k))
$$
($E$ = smooth embeddings, $F =$ functions) where $E(N,X)$ is the fiber over the point of  $F(\Sigma, BO(n-k))$ represented by the normal bundle of $\Sigma$ in $X$.  From this point-of-view we regard the difference between $E(N,X)$ and $E(\Sigma,X)$ as understood and we choose to work instead with $E(N,X)$.


Let $C$ be the closure of the complement of $N$ in $X$. Then there is a fiber sequence
$$
\text{Diff}(C)  \to \text{Diff}(X) \to E(N,X)
$$
where $\text{Diff}(C)$ denotes the diffeomorphisms of $C$ which preserve the boundary point-wise. 

On homotopy groups the last displayed fiber sequence gives a long exact sequence 
$$
\cdots\to \pi_j\text{Diff}(C)  \to \pi_j\text{Diff}(X) \to \pi_jE(N,X) \to  \pi_{j-1}\text{Diff}(C) \to \cdots
$$