Let $M$ be a closed smooth manifold and $E\longrightarrow M$ be a vector bundle with a flat connection $$\nabla:\Gamma(E)\longrightarrow \Gamma(T^{*}M\otimes E).$$ Consider the space of differential forms valued in the vector bundle $E$ $$\Omega^{k}(E):=\Gamma(\Lambda^{k}T^{*}M\otimes E).$$ The flat connection $\nabla$ induce a differential $$d_{\nabla}：\Omega^{k}(E)\longrightarrow\Omega^{k+1}(E)$$ therefore we can define the cohomology valued in vector bundle $$H^{*}(M,E)=H^{*}(\Omega^{*}(E),d_{\nabla}).$$ Given a submanifold $N$ of $M$, the restriction of $E$ on $N$ is also a flat vector bundle, hence we have the cohomology $H^{*}(N,E)$. So is there a Gysin sequence of the cohomology valued in the vector bundle $E$ for the pair of $(M,N)$ ?

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Yes. What you denote $H^*(M,E)$ is actually $H^*(M,E^{\nabla})$, where $E^{\nabla}$ is the locally constant sheaf of (local) horizontal sections of $E$. And there is a Gysin exact sequence for such sheaves, see for instance Dimca's book *Sheaves in Topology*.