People is plural, and cultures is plural. The sentence you have makes sense.

"As many people as possible" refers to how many human beings you wish to include or pack into a single space. "I want to invite as many people as possible to my party." "We want to reach out to as many people as possible." "As many peoples as possible" would refer to the thousands of ethnic and cultural groups and sub-groups which comprise the world population. I have never seen a sentence which stated "as many peoples as possible," but this would be the meaning nevertheless. "Peoples" is more commonly seen in sentences which refer to "the peoples of Sub-Saharan Africa" or, simply, "the peoples of the world."

"People" is generally not pluralized but it can be if you're talking about different groups of people (i.e. Japanese people, American people, German people = different peoples). "Culture" can definitely be pluralized.