This can be done using Wowza media server - just need to buy a license and install. Then you can use build in mechanisms or add your own software billing your clients for using your wowza (a website or a server application).
You need to have an architecture of few servers or buy a plan in Wowza (proper servers can be easily purchased at AWS or google cloud) to ensure SLA.
I'm a software engineer with almost 15y of exp, I'm an expert of streaming, can install all of it for you.
Other architectural options: red5 server, nginx-rtmp + some custom server software (PHP, nodejs, java, python). I can also provide iOS and Android app for streaming (have an app, can add your logo).
Let me know if you are interested, we can talk about the details.
If you need your own software written for you from scratch, this will be rather $80K USD than $5K, because ensuring the streaming software doesn't break and runs on different devices is quite a challenge.