Scraper to grab a Stack Overflow profile name
IMO the answer highly depends on how you are planning to extend the functionality of your class.
If its only function is fetching the username, it's probably better to transform it into a function. Class is an overkill.

A few thoughts if you're going to expand it:


You can take a look at the Stack Exchange API and see if it matches your needs if you haven't already.
You can cache user data in the class fields.
If you don't care too much about making one extra request, you can make one when you call get_profile. For this, you'll probably need to store the session credentials.
You can try using HEAD method instead of GET when you don't need the body (dunno though if the site will handle it as intended).


Also, in Python 3 there is no reason to extend object for new classes.