Possible, sure, anything is possible. But it would require a lot of work.
You can use ArrayMesh to generate geometry (like this tutorial shows how to make a sphere):
https://docs.godotengine.org/en/stable/tutorials/content/procedural_geometry/arraymesh.html
You can also see here for some other options:
https://docs.godotengine.org/en/stable/tutorials/content/procedural_geometry/index.html
However, editing the model would require regenerating the whole mesh, which is likely very slow. There is a function call "surface_update_region" which should allow you do just change a small portion of the model vertices, which I believe would be fast enough for a sculpting program. But I have never used it and it seems it might have some issues (at least in GDScript). With C# it might be faster, and I think it's being overhauled for Godot 4.0. You can read more about it here:
https://github.com/godotengine/godot-docs/issues/3787