Let $f:\mathbb{R}^n\to \mathbb{R}^n$ and $M\in\mathbb{R}^{n\times n}$.
Am I using the chain rule correctly to calculate $$ \nabla_x\langle f(x),M f(x) \rangle= \nabla_u\langle u,M u\rangle |_{u=f(x)} D_xf(x) ~~~?$$
where $D_x$ is the matrix of first derivatives of $f$.
$h(x) = \langle f(x),M f(x) \rangle$ is the composition $h = g \circ f$ where
$$g(u) = \langle u,M u \rangle.$$
The chain rule states that
$$h^\prime(x) = g^\prime(f(x)) \circ f^\prime(x).$$
So yes, what you wrote is correct, knowing that $$g^\prime(u).k = \langle u,M k \rangle + \langle k,M u \rangle$$ which leads to
$$h^\prime(x).k = \langle f(x),M (f^\prime(x).k) \rangle + \langle f^\prime(x).k,M .f(x) \rangle$$