Nope, in Unity you can just click a button and everything happens for you!
Step by step:
in your unity project, go to "File" -> "Build settings" -> select WebGL as platform and click switch platform.
The you can build the game. zip everything in a .zip file and upload it to itch.io. Then you can select "this file will be played in browser". and you're done.
Here is the itch.io WebGL info page: https://itch.io/docs/creators/html5
Good luck!