You cannot manage your peers

I regularly get questions like "hey, $PERSON_X isn't doing what they need and my team is going to look bad as it means we can't do our job. What can I do?"

These people are usually looking for a way to make the person do what they need, but as peers they lack any formal authority over the process.

I push people towards communicating in public, making it clear that you're asking for input/data and no-one's engaging with you. Louise takes this advice and explains it eloquently (emphasis mine):

No leadership means no clear ownership, no visible commitments, no dates. You can introduce all three, not by decree, which you have no power to issue, but by proposal, which you do. “Shall we just write down who is taking which piece and roughly by when, so we stop treading on each other?” is not you managing anyone. It’s a reasonable colleague suggesting the team get organised, and almost nobody will say no to it because everyone is quietly suffering from its absence.

The trick here is that structure does what authority cannot. A peer will ignore you telling them to work harder. They will, however, find it much harder to ignore a shared and visible list, where they’ve publically taken a piece of work with a date on it, next to everyone else who has done the same. You’re making the absence of delivery visible, and letting the visibility apply the pressure - exactly the job the missing manager was supposed to do.

If you're really worried that your team's going to be in the crosshairs for not delivering what you need to within 30 seconds of the team you're depending on shipping their part, there's an easy solution:

keep a neutral, unemotional record of dependencies and what happened, the same way you would document anything. Not to snitch, and not to build a case out of malice, but so that when the situation is eventually looked at, by a new manager (or anyone tbh), the account is accurate rather than whatever settled in the fog. In a vacuum, the person with the clear record of what actually happened is the person who does not get scapegoated.