there are no stakeholders
If there were no stakeholders, the projects wouldn't have existed or had money in the first place. Chances are, they are not properly identified, and this is where you have to start.
It's not like the whole team should go search for random stakeholders. You can have a single person doing that, and, unsurprisingly, it better be a Product Owner.
it's unclear who the actual Product Owner is on the client's side
You don't have PO on a client side (as opposed to a PO on your side). You just have a PO. If the client cannot provide one, then you need to make a proposal. The job of a PO, according to the Scrum guide, is this:
The Product Owner is responsible for maximizing the value of the product and the work of the Development Team. How this is done may vary widely across organizations, Scrum Teams, and individuals.
The Product Owner is the sole person responsible for managing the Product Backlog. Product Backlog management includes:
- Clearly expressing Product Backlog items;
- Ordering the items in the Product Backlog to best achieve goals and missions;
- Optimizing the value of the work the Development Team performs;
- Ensuring that the Product Backlog is visible, transparent, and clear to all, and shows what the Scrum Team will work on next; and,
- Ensuring the Development Team understands items in the Product Backlog to the level needed.
None of these require him to be either client-side or supplier-side.
A best call would be to appoint a person with biggest industry knowledge and best customer communication skills to be your PO.
For example, we have to write user stories and acceptance criteria from the client's perspective. As developers, we are probably not good at this.
Your PO should be accountable for it. Having said that, you are encouraged to add stories and work with PO to specify and refine acceptance criteria. The goal is not to do the requirements gathering work instead of your client, but to have something that you can commit to deliver. This is one of the points that should help with this:
How can we push the client to step in and take more interest in the agile process?
You cannot simply push anyone, unless they are your direct reports (and even that would be unwise). You should show the value of the agile process to the client. For example, reduction of time to market, better feedback cycles, less unpleasant surprises upon delivery, cheaper overall cost of development, etc. There is plenty of material available around that highlights benefits of agile in general, and Scrums, for example, in particular. Slideshare alone has tons of slide decks pursuing the same goal.