The problem is worth studying both when the drawing must be planar (no crossings allowed) and when it is not.
If such drawings exist, then it is also worth studying what grid-size is needed, and whether it can be done with integer coordinates at all. If such drawings do not always exist, NP-hardness should be investigated.
Of particular interest therefore, are planar graphs of treewidth 4 and higher.
The problem was solved negatively in [Fra10]: Theorem: "There exists an infinite class of maximal planar graphs that admit no isosceles planar drawing." Frati raises the new question of whether or not every triangulation admits a possibly nonplanar isosceles drawing.