I inherited this site from a different team and am now being asked to make several changes. I need to edit the links on the top thin blue banner (see screenshot) which is obviously a customization, but I cannot find where to edit these.
You'll need to find out where they're stored and where they're referencing from. I believe you'll be running a custom master page right now, so I'll give you a couple examples on possible situations.
To find the master page, you'll want to go to your root site collection and either edit it in SharePoint Designer and navigate to the master page, or go to the library via the site itself. To find out the master page in use, use the following PowerShell Script:
Add-PSSnapin Microsoft.SharePoint.PowerShell $web = Get-SPWeb <rootSiteUrl> # without the /SitePages/page.aspx or similar $web | Select *MasterUrl
Check out the two links you should get (MasterUrl and CustomMasterUrl)
Hard Coded Links
The easiest way would've been to hard code them and added something similar to
<a href="www.bmo.com">BMO</a> for each of the links somewhere inside the
#s4-ribbonrow container. Change them as you need.
There's probably more options out there, but these seem the most likely to me (bar the last...) ranging from Solutions to server side code, to Page templates that could be edited, as well... It really all depends on what kind of administrators the previous team were. Good luck!