  Using the duality between color and kinematics, we construct two-loop
four-point scattering amplitudes in $\mathcal{N}=2$ super-Yang-Mills (SYM)
theory coupled to $N_f$ fundamental hypermultiplets. Our results are valid in
$D\le 6$ dimensions, where the upper bound corresponds to six-dimensional
chiral $\mathcal{N}=(1,0)$ SYM theory. By exploiting a close connection with
$\mathcal{N}=4$ SYM theory - and, equivalently, six-dimensional
$\mathcal{N}=(1,1)$ SYM theory - we find compact integrands with
four-dimensional external vectors in both the maximally-helicity-violating
(MHV) and all-chiral-vector sectors. Via the double-copy construction
corresponding $D$-dimensional half-maximal supergravity amplitudes with
external graviton multiplets are obtained in the MHV and all-chiral sectors.
Appropriately tuning $N_f$ enables us to consider both pure and matter-coupled
supergravity, with arbitrary numbers of vector multiplets in $D=4$. As a bonus,
we obtain the integrands of the genuinely six-dimensional supergravities with
$\mathcal{N}=(1,1)$ and $\mathcal{N}=(2,0)$ supersymmetry. Finally, we extract
the potential ultraviolet divergence of half-maximal supergravity in
$D=5-2\epsilon$ and show that it non-trivially cancels out as expected.
