No response after client inquiry and I responded?

I've had this happen to me several times. I would send her one more message, asking if she is still interested in having your pet sit, then archive the request. If she answers later on, it will come back to your inbox, so it's not lost forever. There are a myriad of reasons why she didn't answer - she found someone to do it for free, she found another sitter, plans changed, etc. Don't take it personally.