Question: Many may know the conspiracy theory about the world being secretly run by lizard people. In English I also hear the phrase "reptilian shape shifter" associated with the topic, but what would a native Chinese speaker call them? Is it just 蜥蜴人？

Answer: 
I think that 蜥蜴人 is acceptable. I found this entry on Baidu which is usually written in Chinese for Chinese people so if that's how it appears there then at least it makes sense to a significant number of people: <https://baike.baidu.com/item/%E8%9C%A5%E8%9C%B4%E4%BA%BA/8815302>

I would say that a "reptilian shape shifter" isn't the same as a lizard person because there needs to be some distinction between someone who is a 'hyrid' form between a reptile and a human versus someone who can shape shift into that form.
We just say 蜥蜴人.

It is natural for us to call a lizard that have a human liked appearance.
蜥蜴+人 this pattern is like 中國+人(China + people).We say 中國人 ,which means Chinese.

Because we think a lizard that have similar body structure like a human we say 蜥蜴人,which means **lizard+shape like human**. .