I am attempting to create a skybox based on objects in the scene. I can’t be limited by the camera far clipping distance because the objects in the scene are way beyond the farthest possible render distance (in other words, stars). I’m not really experienced in creating textures. So how would I achieve something like this? Or is it even possible?
You can use two cameras,
first camera will be for skybox , you can have sphere or whatever there, set its layer to let say skybox, than in the camera in culling mask set it to show only skybox.
second camera will show anything else, in culling mask select everything except your skybox layer.
Than adjust depth so camera two renders on top (higher depth value).
This should give you required effect.
And regarding your skybox procedural generation you did not provide enough information to answer.