I think yes, as long as the discussion is only at a single point $p$. I think you mean $L:T_p M\to T_q N$ for some points $p$ and $q$, since $L(p)$ doesn't make sense, unless I've misunderstood.
Using charts which send $p$ and $q$ to 0, your question becomes "can I find a map between euclidean spaces, the derivative of which is the linear map $L$?" Since the derivative of a linear map is itself, you can take $f = L: \mathbb R^m \to \mathbb R^n$ (where you use the same chart coordinates to identify the tangent spaces with euclidean spaces) to be the representation of your map in these charts.
Then you can multiply by a bump function which is identically 1 on a smaller neighborhood of $p$ (so as not to affect the derivative at that point), in order to define a function on the whole of $M$.