Let $\nabla$ be a connection on a vector bundle $E \to M$, and let $p$ be an arbitrary point on $M$. Does there exist a local frame of $E$ such that the connection coefficients of $\nabla$ with respect to this frame vanish at $p$?
Such coordinates always exist when $\nabla$ is the Levi–Civita connection on a Riemannian manifold, but I haven't seen any information on the general case.
I think this problem can be reduced to solving the following differential equation. Suppose $(e_i)$ is a local frame of $E$ with connection matrix $\theta = (\theta_{ij})$, i.e. $$ \nabla e_i = \sum_j \theta_{ij}e_j. $$ If $(e_i')$ is another local frame of $E$ with connection matrix $\theta'$ such that $e_i' = \sum_j g_{ij}e_j$, we have the transformation law $$ \theta' = dg \cdot g^{-1} + g \cdot \theta \cdot g^{-1}. $$ Note that each product in this equation is a product of matrices. Thus, in order for $\theta' = 0$, we must solve the differential equation $$ 0 = dg + g \cdot \theta. $$
I figured it out thanks to Deane's comment. The key is that the equation $$ 0 = dg + g \cdot \theta $$ only needs to hold at $p$. (In fact, by the above argument it admits a solution everywhere iff the connection is flat.) We can take $g_{ij}(p) = \delta_{ij}$, so that this equation becomes $$ 0 = dg(p) + \theta(p). $$ That is, we need to find $g$ so that its derivatives at the point $p$ are prescribed by the connection matrix $\theta$. It suffices to take $g$ to be linear in some coordinate system.