Search facility on email inboxes no longer working on my iMAC

What do you mean by the search function? Do you mean spotlight?

Then you might need to rebuild the spotlight index:

1. Choose Apple menu  > System Settings.
2. Open Spotlight settings:

• • macOS Sequoia or later: Click Spotlight in the sidebar, then scroll down and click the Search Privacy button on the right.
  • macOS Sonoma or macOS Ventura: Click Siri & Spotlight in the sidebar, then scroll down and click the Spotlight Privacy button on the right.

1. Drag the disk or folder you want to re-index to the list of locations that Spotlight is prevented from searching in. Or click the add button (+) and select the disk or folder to add.*
2. From the same list of locations, select the disk or folder you've just added. Then click the remove button (–) to remove it from the list so that it can be reindexed.
3. Click Done, then close System Settings. Spotlight will re-index the contents of the disk or folder. This can take some time, depending on the amount of information that is being indexed.