Method 1:

 ans = Transpose[
 Map[Thread,
 D[{f[x, y], g[x, y]}, {{x, y}}] /. {x -> lx, y -> ly},
 2],
 {2, 1, 3}];
 MatrixForm /@ ans

Method 2:

 ans = MapThread[
 Function[{x, y}, Evaluate@D[{f[x, y], g[x, y]}, {{x, y}}]],
 {lx, ly}];

----

*Answer to original question*

Your Jacobian matrix is constant. There's no need to evaluate it at many different points.

 JacobianMatrix[{f[x, y], g[x, y]}, {x, y}]
 (* {{-10, 0}, {0, 8}} *)

Note a common way to compute the Jacobian matrix is (search for Jacobian the docs for [`D`](https://reference.wolfram.com/language/ref/D.html)):


 D[{f[x, y], g[x, y]}, {{x, y}}]