Who has authority over stadium TFR exceptions in VFR flight?
The answer will vary based on the location - San Francisco, Los Angeles, and New York may each have different arrangements. However, in general, it'll come down to how the different ATC facilities have the situation worked out with each other.

The stadium TFR itself does not help much. The relevant section states that:


  THE RESTRICTIONS DESCRIBED ABOVE DO NOT APPLY TO THOSE AIRCRAFT AUTHORIZED BY AND IN CONTACT WITH ATC FOR OPERATIONAL OR SAFETY OF FLIGHT PURPOSES


Any of those facilities could provide the authorization - so it's then down to which will do so. The answer to that will depend on the structure of how the facilities work with each other. The ATC facilities will have Letters of Agreement which state that a given area of airspace is managed by a given facility, as well as how handoffs are handled.

The facility that can provide flight following is also the facility that can authorize you into the TFR in that area. To figure that out, all you have to do is ask: call up Norcal Approach or Oakland Tower, and ask for flight following. If they can do that, then they're the right folks to ask about transiting the TFR. If not, they'll give you another frequency to try. Of course, just because they have authority over the airspace doesn't mean they'll grant the request.