Possibly doable… WiFi and ESP is the predominant solution with Blynk.
Cnnecting Arduino via Ethernet works, but tends to be flakey (usually due to the adapters). This will be hard to do with 10 of them, but it is best to keep the connections from the adapter to the router (and then to the RPi) as direct… going through switches and hubs seems to cause many issues.
For further info, you will need to search this forum for keywords like Ethernet, Wixnet, etc. and start reading.
And experimenting with the sketch options…