This seem to be true, and here is perhaps an over-sophisticated proof. By Poincare duality, homology identifies with compactly supported hypercohomology with coefficients in the Verdier dualizing sheaf $\omega_X$ (resp. $\omega_Y$), which is in the smooth case a shift of the orientation local system. The map in homology corresponds via this identification to the integration map $$H^*_c(X,\omega_X)\to H^*_c(Y,\omega_Y)$$, so we want this integration map to be an isomorphism.
This map is obtained from the counit $f_!f^!\omega_Y \to \omega_Y$ of the adjunction between derived categories $f_!\colon D(X)\leftrightarrows D(Y):f^!$ by taking sections with proper support, so its enough to show that this counit is an isomorphism in $D(Y)$. Here, $D(-)$ is the derived category of sheaves of abelian groups. Since $f$ is a submersion, we have a natural isomorphsm $f^!A\simeq f^*A\otimes \omega_f$ which shows that $f^!$ satisfies base-change along arbitrary smooth map $i\colon Z\to Y$.
Namely, for the projections $i'\colon X\times_Y Z\to Z$ and $f'\colon X\times_Y Z \to X$, we have $i^*f^! \simeq (f')^!(i')^*$.
In particular, taking $Z= \{y\}$ for $y\in Y$ and using also proper base-change, we see that the stalk of the map $f_!f^!\omega_Y\to \omega_Y$ at $y\in Y$ is the (derived) tensor product of the integration map $\int_{X_y}\colon C^*_c(X_y;\omega_{X_y})\to \mathbb{Z}$ for the fiber $X_y$ of $f$ over $Y$ with the identity map of the stalk $(\omega_Y)_y$. Finally, the assumption that all the fibers of $f$ are acyclic implies that the integration map $C^*_c(X_y;\omega_{X_y})\to \mathbb{Z}$ is an isomorphism in $D(Ab)$, by Poincare duality for the acyclic smooth manifold $X_y$.
