Not a sphere, we know that can’t happen due to the topological constraint you brought up. Instead picture a float plane tiled by equilateral triangles. Now mark each vertex as either low, middle, or high such that every triangle has one of each. Push the low ones below the plane and the high vertices above the plane by some amount x. Now it’s a bumpy plane full of triangles, each one is isosceles and all identical.

I think that’s the only way to do this, but maybe there are more. Could we get a hyperbolic plane this way? Normally you squeeze extra triangles around each vertex to do that so I doubt it but maybe.