Can current consultants coach candidates/students online while working at an MBB firm?

No, you can't. 

You can coach and mentor candidates that are in the process of applying to your firm especially if the firm is currently in an effort to recruit new people. But you are doing this as part of your job, not as an extra pay. 

There are actually quite a few restrictions when you join a large consulting firm. For instance, you're going to have to ask for approval every time your want to invest in stocks and usually you are not allowed to invest in individual stocks, but rather in indexes. WIth great power comes great responsibility and also seemingly a restriction in rights. 

(edited)

Hello,

You cannot - I believe there is a clause about this somewhere in the T&Cs when you sign your offer. The reason for this is about the company's image and reputation - as Ian pointed out, you cannot earn money from their brand name while working there directly. Getting paid for recruiting help that is not part of the company's official recruiting process might also be seen in a bad light, as if candidates are paying you for access to the company, which these firms want to avoid. Of course, there are many internal opportunities for consultants to coach and mentor candidates as part of the company's official recruiting process. In any case, the work hours can be long and somewhat unpredictable, which would make it difficult to coach consistently.