Would this type of orbit be possible to achieve without external forces being applied? Is it possible at all? I have little to no background in the sciences so please explain like I'm stupid. Thanks!
This is called the 3-body problem. You can find depicted a number of stable orbits. Here is one in 2d that I like.
There is a collection of 3-body orbits in this bizarre site. http://three-body.ipb.ac.rs/
but the text appears very, very slowly...
That orbit is possible, if unstable (would need active correction from space station).
You would also be interested to Low Energy Transfer Orbits where you can plan a trip between several bodies (planets or, more easily moons) by switching orbit at certain exchange points.
These orbits are "slow" if compared with other solutions, including the classical Hohmann Transfer Orbits, but require very low impulse, perhaps compatible with Space Station attitude engines.