Thanks for looking into this for me. I did reach out to the developer and heard back from him:
Yes, this is a known issue with LocalWP.
The short story is that LocalWP runs in a virtual machine, where the site cannot call services on itself.
And because WP Synchro does that, it fails.
A few guys from LocalWP just contacted me (based on your question on Twitter) and I sent them a long explanation about the issue and how they can fix it.
I just sent it 5 mins ago, so hopefully they will fix it on their end as it might also generate problems with other plugins.
There is no current workaround to get it working, until they either make a fix or I change WP Synchro to work in another way.
If I hear something from them, I will let you know.
I highly enjoyed using Local WP for the most part but I have discovered another local server solution and have made the switch and WP Synchro is working with it.
Had Local WP had a solution, such as Magic Sync as a standalone plugin rather than only working with wpengine or getflywheel hosting I would have gladly paid for it.
If Local WP does find a way to get plugins like WP Synchro working with it, I would be glad to consider moving back.