See my comment above; I think the answer is highly localized to the company. I'm going to suggest two levers that could be used, but either of these could be a career ender in your company.
The core of the answer is that you are responsible for project closure, not completion. If the project is going to fail it is in the company's best interest (and your best interest) to fail fast.
The first lever to use is risk management. Document that:
Project X is three months behind because the product manager left the company. The project staff are being diverted to work on projects where the product manager is involved and active, and I don't blame them. This is 90% likely to delay production at least six months (three months to hire a new product manager and three months for him to get up to speed). We cannot mitigate this risk on our level and it will only get worse as time goes on.
Project Y is two weeks behind, and falling further behind. We cannot begin design until we have a set of testable and deliverable requirements. Product manager YPM has not been able to supply them - he has high level goals, but we haven't been able to pin him down to testable deliverables. If we proceed without requirements, the project will fail unless we receive divine intervention; my staff are good, but they cannot read minds. I'm asking you for permission to restructure the project to include users X,Y, and Z and stakeholders who will own the requirements. My staff will sit down with them and develop a set of requirements that we'll submit to Product Manager YPM. I'd also recommend that we revise the schedule to deliver a prototype - that is a standard way of mitigating the risk of fuzzy requirements. What do you think?"
PMI says that you have the power to close a project unilaterally; that's not true in the real world, but it is still a lever you can use.
- "Boss, Project X will fail in six to nine months. The product owner has left the company and the company hasn't appointed a replacement or even an effective acting product manager. This project doesn't have the stakeholder support required to succeed. Unless you can convinced the board of directors to appoint a product manager, I would recommend closing the project and reassigning the staff to other projects."
- "Boss, Project Y will fail in six to nine months. The project plan identifies "requirements" as the next critical deliverable. I've had three separate meetings with the product manager and he hasn't been able to supply any requirements. He has some high level goals and visions for the project, but no testable, deliverable requirements. I've met with him and told him that I planned to talk with you and recommend cancellation unless we can bridge the gap between goals and requirements. He is unwilling to budge. If this project is important for the company, I think I need some support to set up a summit where I can lock my staff and his staff in a room until they can build a bridge between the goals and requirements."
The key is that in all of these cases (1) you demonstrate that you've done as much as you can, (2) you're asking for specific support, and (3) your recommendations are in the context of the company's bottom line/best interest. But you've got have really done your homework and you've got to have developed a relationship with the boss/stakeholder/whoever that will generate the trust you need.
Of course the final lever is to seek a new job. If you can't address the situation through risk, and you can't address the situation through recommendations to stakeholders, then your best approach may be to follow Product Manger X out the door. But I'd make very sure you have a good way of explaining why you chose that exit strategy for the next interview.