Do I wait for a client to find me on Rovers website and Rover contacts me for the client?
I recommend some form of "marketing" for yourself. On your Dashboard are buttons for posting on social media such as Facebook, Craigslist, etc. You can also get business cards from the Rover Store.
What I did is create an advertising flyer about my services and post copies up at pet stores and dog parks, along with some business cards. I also posted on Craigslist.
However...Rover does do advertising for themselves. When I do a Meet & Greet I ask the client how they found out about me, and so far (for the month since I joined) my clients have heard about Rover from various sources, and have found me by the website search. So much for my flyers! :-D
When I joined I didn't do any outside advertising and it took about a month to get my first request. After that it was another month before I got a second request. But after booking the first couple stays the requests started coming in much more regularly. So if you're wanting to start getting requests pretty quickly, you'll probably have to do some legwork to promote yourself.