May I make a flyer to hand out advertising Rover Services?

You could contact Customer Service directly to ensure you get the right answer ... or you could just have flyers printed that do not contain Rover URL, promo code, or graphic elements and instead provide an email and/or phone number. Then you can measure the response of the flyers and choose to book desired services through Rover (telling them about your rover account, giving them your promo code/URL when you meet) or on your own if you wish.