It will take some time, but this is very easy with the right setup. I'm not sure how much empire ethics matter, but I used Authoritarian/Xenophile/Militarist. Xenophile is probably necessary, the others might not be. You can also unlock "A Home Away From Home" for conquering another species' homeworld through this guide.
Set up a new game with your empire with these galaxy settings:
* Galaxy size: large
* AI empires: 1
* Fallen empires: 0
* Endgame crises: no
* Advanced AI start: 0
* Habitable worlds: 5x
When the game starts, make sure to survey and set up mining and research stations nearby. Build a few more science ships and send them out to survey nearby systems. With the amount of habitable worlds you will eventually find several primitive civilizations. You'll need to build frontier outposts near them and then an observation post (you can also colonize a planet nearby if you want to save on the influence cost of maintaining an outpost, the civilization just has to be within your borders), and when the observation post is built you can begin technological enlightenment of the native species, as long as you have the policy "native enlightenment allowed". This will consume society and energy points each month and how long it takes to enlighten the species depends on how far along they are technologically - a space age civilization will take a very short amount of time compared to a bronze age civilization, for example.
Once the species is enlightened, they will become your protectorate. For them to become a vassal, they need to research 50% of all the technologies you have already researched, so the earlier you can get a species enlightened the better since it will take them less time to complete the research. Just do this three times and wait for them all to vassalize and then the trophy will pop. I got it around 2280 so it will take awhile but it's easier than demanding vassalization from an enemy empire.
When they are all vassalized you can then integrate them into your empire which will give you the trophy for conquering another species' homeworld, and can also help you get "Omnicultural" for having your leader be of a different race than your founding species. Integration takes I think around 20-30 months and costs you some influence but you also gain access to their worlds, tech, and fleet power when it's done.