Just as a follow up, we've determined so far that it seems the new version of the SendGrid plugin could be conflicting with other plugins, but we aren't sure which ones. The Postman plugin is the same, and it appears that this has happened previously as well:
https://wordpress.org/support/topic/postman-smtp-not-working-with-woocommerce
...although past instances may not have involved the SendGrid API.
In one case, one of our hosting clients had a theme bundled with very out of date WooCommerce template files which could have been at fault, but in other cases they are on very modern and well maintained themes, like Avada.
If we can come up with a list of plugins conflicting with SendGrid (either the SendGrid API, or the SendGrid plugin and/or Postman plugin) then we will try to post back here again, although still not clear. Cheers ~
P.S. @justinjwilson yes it seems some sites also have had their API key and/or username/password completely removed from SendGrid plugin settings