Is there any system that enables online office hours?
Yes there are multiple systems; for instance blackboard collaborate (previously Elluminate), which would require your institution to have a license. Blackboard Collaborate is a Web conference tool (video, speech, text, white board, screen sharing etc) which is can be integrated with the Blackboard Learning System. However, it is not necessarily bound to it, and you can get a licences, and use it, without the Blackboard Learning System. You can even get a 30-day trial; http://try.bbcollaborate.com/trial/register.go
More off these systems exists, often bounded to or provided by sellers of Learning Management Systems (LMS), also known as e-Learning Systems. However there are also systems which are either used without LMS (as also is the case with Blacboard Collaborate) and systems which can be used with, for instance, Microsoft Sharepoint (such as Adobe Connect, which can also be used stand alone).
It would be helpful if OP could answer which Learning System (and possible other products) his/her institution uses. As this could indicate what is available. If OP would answer this question this answer will be edited if this would lead to new information.