Is there a time limit for the Supreme Court to either grant or deny certiorari for a case, or can they keep the case in an unresolved legal limbo indefinitely?
There is no time limit. According to this, the average decision time is six weeks from the filing the petition.
As a general process, once a cert petition is filed, there is 30 days for the respondent to file a response and then 10 days for the petitioner to file a reply. Once the petition is briefed, the case is assigned to one of the Court’s “conference” days (I think there is a way to get a copy of the “conference list”, i.e., the cases to be addressed at a particular conference).
At the conference, the petition is generally either granted or denied (generally denied) and that fact is noted on the next “order list” (which usually comes out Monday mornings). The Court can “relist” a case (which means formally putting on a future conference list to be discussed again) or “hold” the case (which means just do nothing). A “relist” is generally a good sign for a petitioner.
The only cert petition I ever filed was denied almost nine weeks after it was filed (but only three days after the conference).
I’m not sure I agree that it’s “legal limbo”. To the extent that granting cert is entirely discretionary, the case is essentially over unless and until the Supreme Court intervenes (and it probably won’t). One area where it might be in “limbo” is if there is a stay pending appeal.