No. You do not have the strain. You have described an out-of-plane displacement on a thin plate or a membrane. That is not how strain is defined. Strain will still be defined in terms of how the membrane stretches, IN-PLANE, as a strain tensor.
Note that the "strain" here will actually have multiple components, and will vary depending in which direction you look, so a strain matrix (tensor), with two principal strains. In order to understand the in-plane stresses and strains, you would want to use tools for modeling the behavior of a thin plate or membrane. Which one it is depends on the relative thickness of the material, compared to the size of the elliptical region. Note that membranes tend to buckle quite easily, so think of a piece of plastic wrap stretched across the top of a bowl.
You can go quite deeply into the mechanical engineering here, or you can probably find some simpler rules laid out for the deflection of a plate under pressure. As you say, there are probably tables or simple formulas written out for the deflection in a 1-d problem. (I never went in that direction in my days as an ME, and I am many years away from that.) There may even be ready made solutions for the case of a circular domain. But to find the solution over an elliptical domain, I'd bet you would need to do some modeling.
In any case, this is not something you will find a simple function for in MATLAB. Yes, you could solve it. But that would probably involve setting up the appropriate differential equations (PDE) over your specific domain, then solving the problem.
