We need to set up a workflow system on our site in which content editors can edit content and publish it IMMEDIATELY, but send a notification to a web designer & copy editor to review the changes and make edits/improvements at their leisure.

Every workflow state I've experimented with makes it so whichever role you choose has to approve or reject content. It's important that our editors be able to publish content immediately, in case the designer & copyeditor can't check it right away. But they need to get notifications so they know to do so.

Is there anything out there that'd allow notifications such as these?

Are you using Evoq?

Yes, Evoq 8 Content basic.

Well, I haven't tried this, but it looks like it's possible.

You can create a custom workflow. Then, with that workflow, edit the "Published" state and check the Notify checkbox. It seems that this would do what you have described.