Python coding style from Java background
Unlike in Java, not everything in Python needs to be a class. In Python we tend to reserve classes for representing things with individual state and common behaviour. If you just want to group some functions together, you can put them in a module.
You have no documentation. What do these functions do and how am I supposed to call them? In Python you should give every public class, function and method a docstring.
Why do you try importing simplejson? This is just the externally maintained version of the built-in json module. You don't seem to use any simplejson-specific feature, so what's the point?
Your code is not portable to Python 3, but this would be easy to fix by putting parentheses around the arguments to print.
This code:

#Check user exists
if user.json()['Data'] is None:
    return None

#Return user
return user.json()['Data']


can be simplified to:

return user.json()['Data']

You have code like this:

VARIABLE = requests.METHOD(api_root + '/accounts/' + account_id + STUFF, headers=self.headers)
VARIABLE.raise_for_status()


in three places. You should make this a function.
It's not clear what the point of this line is:

user_id = str(user_id)


All you ever do with user_id is add it to strings, where it will be converted to a string in any case.
I prefer to use str.format to build up strings, rather than +: the former tends to result in clearer code. For example:

api_root + "/accounts/" + account_id + "/users/" + user_id


would become:

'{}/accounts/{}/users/{}'.format(api_root, account_id, user_id)



Comment on updated question

You have failed to avoid the duplicated code: you have just moved it around.