As Peter pointed out already, "wanna" should not be used when writing because it's ungrammatical. When we speak quickly, we tend to combine words together so that they sound like they're one word, but in writing, they still need to be separated into two. For instance:
wanna, gonna (incorrect) = want to, going to (correct)
kinda, sorta (incorrect) = kind of, sort of (correct)
As for your question, the second one ("I want to take their picture") is correct. "Take them picture" doesn't make sense (unless you're using regional slang) because "them" is not a possessive word.