First of all, in case $f$ is not proper, the sheaf $\mathcal{H}^j = R^j f_!(k_X)$ is not defined as a sheaf associated to a presheaf $$U\mapsto H^j_c(f^{-1}(U),k),$$ since that rule is not a presheaf. Compactly supported cohomology is **covariant** for open inclusions, it is not contravariant (presheaves are contravariant).

Using the corrected definition, typically $\mathcal{H}^j$ is not a local system if $f$ is not proper. In case $f$ is a **proper** submersion of manifolds, then $f_!$ equals $f_*$. The rule that you wrote down for usual cohomology (not compactly supported cohomology) is a presheaf whose associated sheaf equals $R^jf_*(k_X).$ The fact that $R^jf_*(k_X)$ is a local system in the proper case follows from Ehresmann's theorem, for instance.

In the **non-proper** case, one family of counterexamples arises as in my answer to the following MathOverflow question.

Is an affine fibration over an affine space necessarily trivial?

Here is the construction. Let $Y$ equal the complex projective line as a complex manifold. Let $y$ and $y'$ be distinct points of $Y.$ Let $\overline{X}$ equal the compact complex manifold $Y\times Y.$ Let $\overline{f}:\overline{X}\to Y$ equal projection onto the first factor. Let $Z\subset Y\times Y$ equal the union of the following three irreducible closed, complex submanifolds: the diagonal $Z_1=\Delta,$ the constant section $Z_2=Y\times\{y\},$ and the singleton $Z_3=\{(y,y')\}.$ Denote by $Z'$ the union $Z_1\cup Z_2.$ Let $X,$ resp. $X',$ denote the open complement in $\overline{X}$ of $Z,$ resp. of $Z'.$ Let $f:X\to Y$ denote the restriction to $X$ of $\overline{f}.$ This is a holomorphic submersion.

Every fiber of $f$ is a complex manifold that is biholomorphic to $\mathbb{C}^\times.$ The compactly supported cohomology equals the reduced cohomology of the one-point compactification (a "nodal plane cubic"), $$H^0_c(\mathbb{C}^\times,\mathbb{Z}) = 0,\ \ H^1_c(\mathbb{C}^\times,\mathbb{Z}) = \mathbb{Z}, \ \ H^2_c(\mathbb{C}^\times,\mathbb{Z}) = \mathbb{Z}. $$ Restricted over $Y\setminus\{y\},$ the sheaves $\mathcal{H}^j$ on $Y$ are local systems.

Now let $U\subset Y$ be an open disk centered at $y.$ The inclusion $Z_3\subset X'$ with open complement $X$ induces a long exact sequence of cohomology with compact supports. In particular, $H^0_c(Z_3,\mathbb{Z})=\mathbb{Z}$ maps to a nonzero element in $\mathcal{H}^1(U)$ for every $U.$ In particular, the germ of this element in $\mathcal{H}^1_y=H^1_c(X_y,\mathbb{Z})$ is nonzero. However, for every $z\in U\setminus\{y\},$ the image of this element in the stalk $\mathcal{H}^1_z = H^1_c(X_z,\mathbb{Z})$ is zero. Thus, $\mathcal{H}^1$ is not a local system.