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Basically when a saved search is updated send a web notification.

as in <https://developer.mozilla.org/en-US/docs/Web/API/Notifications_API/Using_the_Notifications_API|Notifications API>

But doesn't this need to be run client side?

Yes, though im not sure what else you meant by web notification

<https://developer.mozilla.org/en-US/docs/Web/Progressive_web_apps/Re-engageable_Notifications_Push>

something like this but I need to be able to host the server at a different location. I'm not sure if that is possible.

And then send a notification from <http://netsuite.com|netsuite.com> on push  events.  I don't know the protocol well enough to know if I'm making sense.

Dont know the push api, looks experimental