Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of the four vertices are pulled upwards and inwards equally, at the same time?

If I understand the question properly, the answer depends on the flexibility of the sheet.
A fine cloth napkin might crease almost like origami, in which case the height would be half the diagonal of the square,
$\sqrt{2}/2$ for a unit square:


