For a single electron, kind of. There’s no force acting on it outside of the attractive force between it and the nucleus, so it would just end up in an elliptical orbit, and if you just looked at the distance from center, it would appear to oscillate like a spring as it goes around.

This is misleading, though, because springs push and pull in both directions, and that’s not what’s happening here. What’s actually happening is something like Pluto’s orbit, where it’s closer than Neptune half the time and farther the other half, but nothing’s really making it do that besides geometry.

If you have other electrons, like in most atoms, now you get a repulsive force from the electrons, which are all in their own spherical shells, and they interfere with elliptical orbits enough that they eventually become spherical. This would look more like a spring eventually settling if you just look at distance from center.