Learn Spanish (Latin America): Supongo que ellos viven aquí.

In Spanish, the connecting word que (that) is completely required when linking a reporting verb like supongo (I suppose) to the rest of the sentence. While in English we often drop the word "that" ("I suppose that they live here"), you can never skip que in Spanish.

Why is the verb supongo instead of "supono"?

Suponer (to suppose) belongs to the same family as poner (to put). Just like the "I" form of poner is the irregular pongo, the "I" form of suponer is supongo.

Since "I suppose" means I'm not entirely sure, do I need a special verb form for viven?

Not here! Even though you are guessing, verbs like suponer (to suppose), creer (to believe), and pensar (to think) are followed by the normal present tense when making a positive statement. So, you just use viven, exactly as if it were a proven fact.

Could I leave out the word ellos and just say Supongo que viven aquí?

Yes, absolutely! Since viven already tells us that the subject is "they" (or "you all"), the word ellos is optional. It is only included here for emphasis or to make it perfectly clear who you are talking about.