Consider $E$ rank $k$ flat vector bundle over $M$ with locally constant trivialization $U_i$. Define $\Omega^p(M,E)$ as compatible local sections $\Omega^p(M,E)=\Omega^p(M)\otimes_{C^\infty(M)} E$. Assume $E$ has two charts $U\times R^k$ and $V\times R^k$ to obtain locally constant trivialization. Then consider the following sequence.
$0\to\Omega^\star(M,E)\to\Omega^\star(U,E)\oplus\Omega^\star(V,E)\to\Omega^\star(U\cap V,E)\to 0$
The first map is just standard restriction and trivially injection. The second map is given by $(\alpha,\beta)\to(\phi_{UV}\alpha-\beta)$ where constant matrix $\phi_{UV}$ is the corresponding linear map between $R^k$ factors of $U\cap V\times R^k$ and $V\cap U\times R^k$ of $E$.
Consider partition of unity $\rho_U,\rho_V$ subordinate to covering $U,V$. Let $\omega\in\Omega^\star(U\cap V,E)$. Consider $(\rho_V \phi_{UV}^{-1}\omega,-\rho_U\omega)$. Note that $g_{UV}$ are constants acting only on $E$'s vector components. So $\phi_{UV}\rho_V \phi_{UV}^{-1}\omega+\rho_U\omega=\rho_V\omega+\rho_U\omega=\omega$. Hence, the sequence is exact.
$\textbf{Q:}$ It seems that I can induce Mayer-Vietoris over forms with values in flat vector bundle. Is this correct?
Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 7.