Learn Italian: Questi mucchi di vestiti ingombrano la stanza. - These piles of clothes clutter the room.

Mucchio ends in -io, and because the i is not stressed (the spoken stress is on the u), the plural just drops the final -o rather than adding a second -i. Therefore, it becomes mucchi.

Since our current lesson is about the relative pronoun che, how would I say "The room that these piles of clothes have cluttered"? Does the past participle agree?

You would typically say La stanza che questi mucchi di vestiti hanno ingombrato. When che acts as a direct object (representing the feminine singular la stanza), making the past participle agree with it is completely optional. You can say hanno ingombrata, but leaving it as ingombrato is much more common in modern spoken Italian.

Is ingombrare the best verb to use for "cluttering"?

Yes, ingombrare is the perfect verb for taking up space, obstructing, or cluttering an area. At this level, you might also find its noun form highly useful: l'ingombro means a bulk, obstruction, or physical clutter.

Why is it just di vestiti instead of dei vestiti?