I am just wondering if it would be rude for me to ask the client after I am done with the booking to see if they would leave a review
Nope you should ask. Of course, Rover sends the client an email asking them to review you, but many clients don't respond to the request. Many sitters will text the client after the booking and offer a discount if the client books with you again and has left a review, hey even a few bucks off it worth the review.
Don't worry --it is not rude at all.
I usually say something like: Hi Mary, I was wondering if you would leave a review of Kitty's stay when you have some time. Thank you very much, we really appreciate it.
I'm going to try giving the owners a little bag with treats that says it was a "treat" watching (blank) , if you could leave me a review that would be "pawsome". Hope it works...