Mitrall was a Guardian Deity of the Dead, who would watch over the Dead and their tombs until the time of their return (physical or re-incarnation). The culture that worshiped Mitrall has been conquered and destroyed a few centuries ago. Yet the Cult of Mitrall continues.
For generation after generation, the cult has continued. They seldom actually worship Mitrall, they just continue his duty of safeguarding the dead. They recruit new members from those who have had their dead abused or desecrated. (Sometimes they recruit scholars of history, who might help them protect the past (and the dead that reflects it)). The Cult watches over graveyards in the citys and tombs in other places. The Cult goes after those who loot graves or tombs of any nature. This means they often go after criminals and adventurers. They will retrieve any stolen good and monies from a tomb, to return it to the appropriate place. If the tomb was desecrated or the violators did not show some decorum, they will often be killed. In addition, the Mitrall Cult will try to free any bound spirit (that was once a person) and any remains of the dead (so they will attack or arrange for others to attack necromancers of the zombie making persussion). They have taken down a number of Demon Cults which trafficed in bound souls. The Cult of Mitrall is very serious about protecting the Dead so they can re-incarnate or return.
The Cult is quite expert at finding and eliminating their chosen targets. They will use magics and proper detection work to find them. They have inventories of many tombs. They will either take hand in matters directly, or contract out to other appropriate forces to eliminate the target. In one cunning plan, they had a looting set of adventurers attack a necromancer. They then swooped down and eliminated the battle worn survivor. Some adventurers are actually members of the Cults who are moles, set up to sabotage parties invading parties. They have had centuries to perfect these techniques (after borrowing quite a few from Evil Demon Cults).
Additional Ideas (1)
To clarify some points
* "Mitrall was a Guardian Deity of the Dead ..." Is there a sense that the deity doesn't exist anymore, or is this from a campaign that doesn't have interventionist deities?
Non to minimal interventionist campaigns is the ideal for this group. A source of divine spells maybe if that fits the setting, but not even that is required. This way the cult can be used in low fantasy, sword and sorcery, Asian fantasy, modern pulp campaigns, and a few other fantasy variations.
In a campaign with active divine intervention, this cult would be in active conflict with other deities/ religions. Those gods wish older items to be destroyed or stolen, Mitrall wishes to stop it. Mitrall is an ancient god not of their pantheon; it is a risk to their position, As a second point, in worlds with active and easy to access magics, finding people who steal things and finding people who kill other people and finding things that are stolen, are easy tasks. (unless obscured by magics). Thus making the entire point moot.
* Obviously the Cult is very strong, if it has the resources claimed. Is this organization particularly prestigious in its society, or does it have particularly strong backers? Is its scope local or international?
The scope is what you need it to be for your campaign. I would see this as a group international in scope, or at least strongest in the regions where there are a large number of tombs and such for them to protect. It depends if you want them as a vast organization that will thwart player’s raiding tombs or just a local annoyance to anyone breaking into tombs in this one region (the one of course players are going to need to do this). Locally it is probably a group of operatives who watch the tombs or places where people buy supplies to raid OR the inns looking for travelling raiding types. A few are probably warriors, infact most of them are warriors or assassin types. There are probably a few that deal with antiquities and a few scholarly type. As you get older, you probably move up the ladder.
* If the worship of Mitrall is no longer an aspect of the Cult, does the organization still call itself a "Cult," as opposed to a society/sodality/order?
Semantically, it is a cult because of this. Otherwise it would be a sect or religion. They still have a belief in the God and its mission; they just don't spend much time actually doing worship. So no Friday night temple for them; they have a more practical faith.
* Plainly it's well-organized ... but how? Is there a hierarchy? A cell structure? Is it a secret society, or are some of its workings public?
It is a somewhat secret society, as they do hunt down and kill people who violate tombs. This keeps people's relatives from taking vengeance on them. I would of thought that was obvious from the write up. It is hard to be a public group that has secret missions.
Again, what organizational you need is the right one. A modern terror cell structure would of been unheard of (and undreamt of) in a feudal society. The organizational pattern that fits your world is the one you should use. If you have a feudal or noble system, you would probably have a hierarchical system with a strong local leader who has fealty to the higher leaders. Spy/cell structure didn't come into existence in China until the 1200s and Europe until the Renaissance Periods. (Or so I have read recently) So what fits for you and your world is what you should use.