A different Analytic Account can be set on each line of a Purchase Order.
You can already:
1 - click at the bottom of the Vendor Bill in the system notification where you see This vendor bill has been created from: POXXXXX to open the Purchase Order.
2 - from the Purchase Order, see the different Analytic Account(s) used, for each product/service ordered.
3 - click open each Analytic Account and click the Project (or Projects if there are multiple) to see not only who the current PM is, but also any prior PM's that may need to approve the Bill.
[extra credit] - click the Avatar of the PM to see if they are even in the office this week or online right now.
Once you know who is the best person to ask, click once at the top on the Vendor Bill (link in the breadcrumbs) to schedule an approval via an Activity on the Bill (perhaps create a custom Activity "Approval" if you need one).
The request will be emailed to them and they will see it in their Activities dropdown (clock) at the top of their Odoo screen. You will have a record of who approved it and when.
See also:
https://www.odoo.com/forum/help-1/discuss-odoo-14-how-can-we-see-who-is-online-or-offline-in-the-chat-window-like-messenger-fb-194937
https://www.odoo.com/documentation/15.0/applications/productivity/discuss/overview/plan_activities.html
Note: If the PM placed the Purchase Order, you can also just look in the chatter after the first click and see them there / click their Avatar there. You could also add a custom field to the Purchase Order form view so a PM need to be selected when the PO is created, and that would also meant you are just a single click away from discovering who you need to ask.
