If you can do it for rational coordinates, then by a dilation (multiplying the coordinates of the vertices by the l.c.m. of their denominators) you can do it for integer coordinates. On the other hand, if it's impossible for rational coordinates then obviously it's impossible with integer coordinates.

Now suppose it can be done with rational coordinates. Then all three sides of the triangle must have rational slope. But if one side has slope t, then (from the formula for tan(θ+120°)) one of the other sides will have slope

which is irrational. Therefore no such triangle exists.