I've heard that every annulment must go through the Vatican in some capacity. Does every annulment get reviewed by some tribunal there or is it just a rubber stamp for the tribunal of the diocese from whence it came?
My understanding has always been that this is handled by a Diocese's Marriage Tribunal.
On looking it up, it appears that when you request a Declaration of Nullity in Canada, an investigation and judgement are made by the Diocesan Marriage Tribunal, then all decisions are reviewed by a Canadian Appeal Tribunal, which issues the final Declaration (or not.)
This pamphlet (pdf) from the Toronto Archdiocese has a good explanation.
In 2011 when the question was asked in annulment process two courts were needed. The diocesan court investigates the case and comes to a decision. If it thinks the marriage is invalid, the appeal tribunal (normally on metripolitan level) has to decide the same case. If it agrees the marriage is annulled, if not the Roman Rota has to decide.
In 2015 Pope Francis reformed this process with the apostolic letter motu proprio Mitis Iudex Dominus Iesus. Now only the decision of the diocesan court is needed. Of course the defender of the bond and the parties can appeal to the metropolitian court and after that to the Roman Rota.
So Vatican plays no role in the most annulment processes.
Maybe you have heared of Catholic real divorce processes: in favorem fidei (privilegium petrinum) and inconsummation. Both processes end with a signature of the pope (cann. 1142 and 1698 § 2 CIC).