The player shouldn't have to roll to use a background feature
Backgrounds generally don't require you to roll anything in order to take advantage of their feature. They just provide some sort of benefit which is always available to the character (unless the circumstances are such that it's simply not reasonable anymore).
In the example you propose, the character themselves isn't really doing any of the work if they use this feature. They're sending a message to their trusty contact, and that contact is then doing the legwork of finding out if there's any useful information to be had about the subject in the criminal underworld. As the DM, you could decide arbitrarily whether or not they know anything about it, or you might roll on behalf of the contact in order to represent their luck in investigating the matter.
Personally, I would be inclined to treat this feature as the start of an investigation. Perhaps the contact doesn't know anything about the cult personally, but they do know that a local gang of thieves has been seen meeting with cultists - and they can arrange a meeting with the thieves' leader for the character to investigate further. Use the feature to give the party leads to follow up on, but don't just let it shortcut doing any proper investigatory work.
Generally, you'd use Persuasion/Deception/Intimidation (as appropriate) when attempting to convince other people to tell you useful information, and you'd use Investigation when deducing conclusions from the clues you have and physical evidence.