Does it matter what hosting service I use if I want to build a website through the Coaches Console?

The website you build through the Coaches Console is already hosted by the Coaches Console as a sub-domain of main Console website (https://your-prefix.coachesconsole.com). If you want to point an external domain of your own (my-domain.com) to your Console website, it does not matter who provides it (what provider and host you purchase it from).