Can you derive a nonlinear elliptical function describing an orbital
trajectory which best fits two points on the ellipse knowing only the
x,y coordinates of the two points in the orbital plane. If so what
equation describes it?

No, that is an underdetermined problem.  Even assuming you know the
location of one focus (i.e. you know you have two points on a planet's
orbit and you know the location of the sun) there are still three
degrees of freedom.  You can think of these as distance from one focus
to the other, direction from one focus to the other, and eccentricity
of the ellipse.  Or alternatively, length of major axis, length of
minor axis, and orientation of major axis.  Any way you cut it, you
can't turn 2 pieces of information into 3.