Let $M$ be a smooth manifold and $\nabla$ be a covariant derivative on its tangent bundle. The question is that if $\nabla$ is flat then is there any locally coordinate system on $M$ such that in this coordinate the christoffel symbols are zero? In the case that $\nabla$ is the Levi-Civita connection of a Riemannian manifold, then the answer is yes.
A necessary and sufficient condition for the existence of local coordinates in which the Christoffel symbols of $\nabla$ vanish is that both the curvature and the torsion of $\nabla$ are zero. To see that this is neccesary, just look at the formulas for the torsion and the curvature in local coordinates in terms of the Christoffel symbols. The sufficiency is proven here.