What is the base word for refugee?

"Refugee" is a noun, but it has no adverb form. This is the most straightforward answer you can get.

As a guide, you can try to put the Word into a sentence (in the position of an adverb or adjective) and see if it fits in.

In this case, if you can say "He is refugeeish" (meaning "he behaves like a refugee"), then the Word "refugee" has an adjective form, and if you can say "He talks refugeeishly" (meaning "he talks like a refugee"), then the Word has an adverb form. For both cases, you can't do it, so you can safely conclude that refugee is a noun that has no adjective or adverb form.