I have a router that allows only 10 devices to be connected simultaneously, I have 22 led controller in my room.
So I can’t connect them all to the router because the router allows only 10 devices, If I changed the Esphome configurations to allow MQTT will solve this issue? or still I need to connect the controllers to the WIFI to be able to connect to MQTT server?
Yes, you do. Buy a new router 
@Burningstone I liked your straight forward answer 
, Thank you
How is it possible you have a router that only lets you connect 20 WiFi clients in total simultaneously? I can’t fathom how such a thing is possible, unless we’re talking truly ancient hardware?
It’s not ancient hardware, but I got a “temporarily full” message when I try to connect over 10 devices, BTW I bought a new TP-Link Archer C1200 router.
That seems really weird, for anything approaching a modern router. Or someone is trying to do massive segmenting and trying to charge more for what really should be normal capabilities.
Anyway, TP-Link Archer series is a good buy, should work well for you.