Yes, Thorne will grant them some gold to do this operation. They'll need to explain what their plan is, but they'll be given permission and can sign out an allocated sum.
Once they reach the alleyway, they'll find that there is unexpectedly a fresh trail of blood. It's possible this isn't related to the disappearance as it appears very new, but they may conclude that this could be part of a new abduction attempt. They can also follow the trail to make sure no one is in trouble regardless of whether or not it helps their current investigation.
They can, of course, ignore the blood and go inside the gambling hall to question patrons, split up, report back to Thorne, or otherwise approach this how best suits your plans.
no subject
Once they reach the alleyway, they'll find that there is unexpectedly a fresh trail of blood. It's possible this isn't related to the disappearance as it appears very new, but they may conclude that this could be part of a new abduction attempt. They can also follow the trail to make sure no one is in trouble regardless of whether or not it helps their current investigation.
They can, of course, ignore the blood and go inside the gambling hall to question patrons, split up, report back to Thorne, or otherwise approach this how best suits your plans.