When I tried to send my link and coupon code to someone that had inquired about me dog-sitting for them, Rover wouldn't let me send the code
Your code is meant to attract new users to the site. It's not intended for use by clients who have already discovered the website, so there are filters in place which prevent you from sending your code through Rover's messaging system. You can get around it by changing the formatting (so I can say something like "enter the code CITY DOGS 20 without the spaces,") and it will get past the filters, but I usually refer new clients to my facebook page, where my code is prominently displayed. That way if they want to check it out, they get the reward of my coupon code, but I don't openly advertise the code to clients who contact me through the website.
Have you tried to use the Email Your Code option from your Dashboard? I seem to recall another sitter stating they couldn't send via inbox messages, but could through that way.