The Ob basically cares about secrecy first, and its goals second. It's been around long enough that it's in no rush to finish, so it should be straight-up impossible to get the whole story before adventure 7 when the convocation happens.
So, how might a PC join their side?
In adventure 1, seeing all the weird stuff on the Island might prompt curiosity, but there's no connection to anyone actually in the Ob.
In adventure 2, Cillian Creed might manage to recruit a PC, or someone might talk to Macbannin in private, or maybe even witness Nicodemus possessing Macbannin in his cell after he's caught. Getting an RHC constable working for them would be useful, but that would just be Tier 6 at first. "All we want is information. Your mission is to copy all your mission reports and leave that copy in a dead-drop in location X. We'll send a small payment to you in thanks. If you do good work, in time we might give you a chance to do more for us."
In adventure 3, Caius might be convinced to trust one PC a smidge, if that PC could arrange to free some of Macbannin's workers who were arrested previously, and get them passage out of the country to Drakr. They wouldn't actually know anything useful, but it's more of a test of willingness to collude. He'd probably get killed by Sijhen before he could really cultivate the PC, but someone on his payroll might - while the PC is away - drop off a letter at the PCs' home. It's encoded, with a note, "The cipher is OBG," which is a reference to Old Bergeron Gold, the inscription on Caius's ring. If the PC figures out the cipher and decodes the letter, it's a message explaining that Saxby was working with Macbannin and covering up his activities. It invites the PC to meet a contact at a specific shop in Parity Lake if he is willing to help ensure Saxby dies before her conscience gets the better of her and she spills any of this information. If the PC goes along with this, he meets one of the Obscurati shadow mages who is going to lead the Kell guild guys into RHC HQ in the third act. Give the PC some task, like removing a small ward in the basement that's intended to prevent extradimensional portals there. (I mean, it *is* a jail. They'd want to keep people from teleporting out.)
In truth, the Ob know that Saxby doesn't really have any useful information, since her only real link in the hierarchy was upward to Macbannin. If the PC takes this info and uses it to capture Saxby, there's no real loss to the Ob, and it shows the PC isn't really on their side. If the PC does help kill Saxby, word of that gets filtered up various channels, and eventually reaches Lya Jierre.
In adventure 4, the train mission gives the PC the most opportunities to make friendly contact with Ob operatives. Brianne and Ottavia wouldn't know who the PC is, but there are several situations where a PC might help those Ob agents avoid being spied on.
Maybe if a PC gets close to one of the meetings along the railroad, you could pull the player aside and run a brief scene where Ottavia figures out they might be turned into an asset. She just asks the PC to be on the lookout for a signal (like, Bree makes a point to eat some Brie cheese), which means she needs the rest of the party distracted for a few minutes. Ottavia could then use this when she tries to lure the PCs to the cursed island of Odiem, in order to keep them from realizing Luc is still on the train. She then leads them to a trap, but has arranged a boat to rescue her and the PC in the morning.
If the PC agrees, she offers to arrange a meeting with her contact. Then in Vendricce there'd be a chance to do some spycraft -- meet here, go there, take this tunnel that splits in order to shake pursuit -- and eventually meet Lya Jierre just before she heads to the rail depot to meet Luc. Lya would try to judge the PC's actions so far, and would pose the question, "What needs to change in this world?"
If the response is suitably academic, or if the PC has been really helpful, Lya would tell the PC to come along, but to be ready to 'turn against' her if the other constables show up. Her mission for the PC is to be a double agent. Since obviously the ring code has been breached, she won't give the PC a ring, but would hand over a gold coin which is treated with a glow-in-the-dark chemical that reads, cheekily, "Overt Betrayal? Avoid!"
Then she'd concoct a scenario where the PC could reasonably rejoin the rest of the party (perhaps after getting a hand sliced off).
In adventure 5, you'd want to come up with some Ob contact person between the PC and Lya - how about an orc immigrant hair stylist? I dunno, something fun. That contact person says that the Ob are done using Lorcan Kell and guarantee the scoundrel will be brought to justice, but they want for that to wait until after the visit by Sovereign Han Jierre. So the job is to delay.
You, of course, don't want to make the PCs fail, but give the turncoat a chance to betray them. Or stay undercover, stymying them, until adventure 7.
More when I'm back from lunch.