Hi Shelley,
I believe you are really close here, you might have just grabbed the wrong Note "Object" from the left-hand side of the screen designer. To clarify, the Vendor Table has a field labelled "Notes", this would be the value you require, and can be found on the Report Designer screen on the left-hand side, under "Add Fields". If you look at the next tab over form "Add Fields", you will see "Add Objects", this sounds like what you choose instead. There is an object labelled "Help/Note", however this isn't the same as the "Note" field stored in the Vendors table.
If I am correct, simply delete the Note/Help Object from your customized screen and drag over the "Note" field from the "Add Fields" tab on the left, this should fix your issue. If you were to test this, you should be able to add a note after selecting a Vendor, then after pressing the "Save" button (which will "Save all Sections"), you should now have the Vendor Note record updated in Method.
Please give this a try and let me know if this resolved your issue.
-Ben
Screenshot of Screen Designer Notes: