Is this Possible?

I'm new to arduino but I was thinking about a new project.

Would it be possible to hook an arduino up to an esp8266 for Wifi and then use that to stream audio from a youtube/vimeo link?

And for the arduino to get the link it'll host a basic website that has a textbox and a button to say "submit link"

In general, an AVR-based Arduino (like an Uno) is not cable of "streaming audio."
An esp8266 might be able to do it on its own...