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 $$ \cdots\to \pi_j\text{Diff}(C) \to \pi_j\text{Diff}(X) \to \pi_jE(N,X) \to \pi_{j-1}\text{Diff}(C) \to \cdots $$