Assuming we want to ask friends for help, my first reaction on seeing this was that it's useful, but on second thoughts, I'm not so sure.
You probably get this question a lot, but if you're just using Facebook, why can't I post the message myself on Facebook instead of going via you? Or as we used to do, just call or email them?
If the value you are adding is management of the arrangement info, then I'm not sure how useful that is. Facebook messaging or email is good enough for doing that and fits in with everyone's usual mode of communication.
I know this is an MVP and I'm sure you will add more useful features, but you really need to start with the most useful feature, otherwise it will be hard to get people onboard.
Helping people find good professional caretakers would probably be much more useful and will give you a direct way to make money via referrals.
If you are really seeing a lot of interest on the part of non-pet-owners to take care of pets, then this could indeed be an AirBNB-type opportunity - empowering non-professionals to become sellers on a new marketplace. Then, trust would be an issue and there may be other legal hurdles, but those are signals for opportunity.
Edit: One way to establish trust is to enlist people who live in my neighbourhood. The person we currently hire lives on our street, which has increased my trust for them a huge amount. When I can literally walk past their door every time I go out, I feel more comfortable hiring them. You could use online local forums to gauge interest and get people to sign up.