This answer is related to the remarkable answer for a similar problem. The only difference is that the OP asked about monotonicity of a polygonal chain, but not about monotonicity of a polygon.
The algorithm idea is the same - we need to determine ranges of "authorized" angles for normals to $L$ looking at the adjacent chain segments. An angle is considered authorized if a normal with this angle defines such a straight line $L$, that all the straight lines, orthogonal to $L$, intersect the chain not more than once. In case of a polygon it was enough to look at concave vertices only, but in our case we must look at all the internal vertices.
The picture below shows a polygonal chain with four segments. Authorized angles at each internal vertex are marked by green color.
If an intersection of all the $(n-1)$ ranges of authorized angles isn't empty, then the polygonal chain is monotone, and the line $L$ exists. This algorithm apparently will take $O(n)$ steps.