After the initial registration process, there is no further permission required to use Facebook's API. If you reach 5M+ monthly active users/100M+ API calls per day/50M+ impressions per day, additional terms may apply.
As per the Facebook Platform Policy, you must meet several quality criteria:
- Your app must not receive too many negative reviews
- You app must be stable and easily navigable
- You may not alter their logo, imply partnership with Facebook, etc. as per their Advertising Policy
There are of course all the other rules of software that still apply (no using your client to distribute malware, etc.) but to answer your question, yes, you can build a social media client as long as it functions as intended and does not imply a relationship with the social media company.