Composition with a continuous function

Setup. Since $f$ is bounded, $f([a,b])\subseteq[-M,M]$ for some $M>0$; $\Phi$ is uniformly continuous on $[-M,M]$.

Approximation. For $\varepsilon>0$, choose step functions $\varphi\le f\le\psi$ with $\int(\psi-\varphi)<\delta$, where $\delta$ is chosen so that $|\Phi(s)-\Phi(t)|<\varepsilon'$ whenever $|s-t|<\delta$ on $[-M,M]$.

Step functions. Then $\Phi\circ\varphi\le\Phi\circ f\le\Phi\circ\psi$ (where $\Phi$ is increasing — WLOG, working with $\Phi$ is more subtle; but for uniformly continuous $\Phi$, one uses $\Phi\circ\varphi$ and $\Phi\circ\psi$ plus bounds). The integral difference is controlled by $\varepsilon'$ and the length of $[a,b]$.

Conclusion. $\Phi\circ f$ is Riemann integrable.

Remark. The result is standard: composition of Riemann integrable with continuous preserves integrability.