The Vatican miraculously survived the Atomic wars, neither side in the Great War apparently willing to nuke the seat of Catholicism.
While it possesses its own Judge-Inquisitor force - who wielded Absolver guns and whose equivalent of 99 Code Red was Code 666 Red - command rests with cardinals and the Pope. Various internal factions have vied for influence.
In the 22nd century, it had a remit to provide a global defence force against demonic incursions; to this end it has a network of off-world spies, special/covert ops groups named DeepCore and White Ops, and freelancers like Devlin Waugh. Despite its holy practices, it made use of agents with extremely diverse backgrounds. Agents like Waugh had papal immunity from Mega-City One's border restrictions.
Civilians lived around the Vatican, some of them worshipping pagan deities like Cicciolina to free them from papal bondage and chastity. Clones were considered soulless and not to be done by the Pope.
Threats it sent agents against included the Anti-Pope, a frequent for of Devlin Waugh, and the monstrous Herod that devastated Bahrain. The Herod's skull was locked in the Vatican's Iso-vault Zero.
The Vatican was dubious about whether a time-travelling necromancer had caused Judgement Day; this was felt to be a Mega-City One cover story, and the crisis had been divine retribution on the sinful world. Nuking the five zombie-overrun megacities was seen as an obscenity.  They refused to even call it "Judgement Day" as it wasn't the real religious event and instead called it the Apocalypse War (with another fancy word for the Apocalypse War).
In 2116, the Vatican was one of multiple powers trying to secure Judge Eckhart at the Antarctic after he claimed to have contacted God. The Vatican considered the other megacities to be heretics and sent the notoriously sadistic Judge-Inquisitor Cesare as their agent. Cesare was able to recruit the catholic Murphyville Judge Wilde as an ally but both were killed by Judge Dredd.
The Vatican faced a a global crisis in 2122 sparked by the insane Lord Benjamin Hekt, who wanted to summon the ancient alien Nommo from Sirius to transform manking. The crisis started when occult criminal Lady Fading Light had the Vatican's relic vaults invaded and looted. After discovering the Herod's skull was taken, agents Lazlo Fuerst and Lucy Melmouth brought Waugh back into service in the hope to tracking it down. While the Vatican's forces raided Fading Light's occult auction at Castle Neuschwanstein, they were unable to stop Hekt's agent Jack of Mice resurrecting the Herod using his own dying blood; after Waugh fled the mission, Vatican DeepCore failed to stop the Herod from going around the world murdering people born on the 23rd July and were caught out when Jack of Knives used a spirit bomb to trigger the Centre of Pestilence near Indo-City.
An alliance of supernatural villains - called the League of Shadows - including the Catechist and Harry Kiri approached the Vatican to deal with the mutual threat, and dragged Waugh back in. The Vatican discovered Hekt's HekTek satellites had caused a first strike on the League and realised he was the threat: he was triggering Centres of Pestilence to light the way for the Nommo and was going to terraform the planet by flipping the magnetic poles. An attempted Vatican strike at Hekt failed, as he'd already summoned a Nommo guardian and trounced their forces. As the 23rd of July neared, the Vatican had failed to stop the murder of everyone born on the 23rd July or the triggering of every Centre outside of the Arctic.
Vatican's White Ops went in full force to the North Pole, hoping to intercept Hekt while Psi-Tek at DeepCore monitored them. While Devlin and his allies raided Hekt's underwater base, White Ops and the Judges held off the Herod - or rather, tried to as it swatted them side. The mission was yet another failure, with only the last-second sacrifice of Eddie Whiteman disrupting the Nommo's arrival and transformation of the world. It would take months for the Earth's spiritual grid to finally settle. 
After an intelligence failure, Vatican spies in outer space began to be killed off by occult forces in 2123.
In 2126, the secret society Sanctus Dominus, including the Judge-Inquisitor Sextus and inquisition leader Monsignor Cesare, had come to believe Grud had died creating the world. They began murdering psychics and seers and burning their 'souls' out to make prepare the way for the Grudhead Assumed, a mad ritual to recreate Grud under their 'advisement'. Sextus tracked Devlin Waugh to Mega-City One to use his body to take the 'holy' energies but Waugh quietly tipped off the Judges in the hope of gaining aid; as a result, while Sextus was able to pull diplomatic strings to take Waugh back, Dredd came with him. The conspiracy kept Dredd 'helping with inquiries' while they prepared Waugh for the ritual, but the two were able to get together and Dredd disrupted the ritual, making it just temporarily power up Waugh and allow him to slaughter Sanctus Dominus. Unknown to Dredd and Waugh, the papacy was quite aware of the plan and simply went back to trying to recreate the Grudhead. 
- The Cicciolina cult is a reference to the porn star Ilona Staller, who used Cicciolina as her stage name.
- Both Crusade and the Devlin Waugh strips have Judge-Inquisitors but they have wildly different looks: the former has them as sinister black clad figure with ornate helmets, the latter uses a Judge version of the Swiss Guard.