I need to have a list of bills of a vendor that relates each bill to the customer involved

Yes. In QuickBooks Online (QBO) you can generate a report that shows your vendor bills alongside their related customer/project by customizing the Transaction Detail by Account report. 

Here’s how:

1. Go to Reports > Transaction Detail by Account.
2. Click Customize in the top-right corner.
3. Expand the Rows/Columns section and check the box for Customer to add it as a column.
4. Under Filter, select Transaction type, then choose Bill from the Value drop-down.
5. Once done, click Apply changes.

You can also read this article to learn more about customizing reports in QBO: Customize reports.

Feel free to reply below if you have any concerns about reports in QBO.