CiviCRM for animal rescue charity
Look into CiviCase for planning and tracking each animal's adoption appointments, vet visits, groomings, etc.

CiviEvent has integrated location/venue components that might be useful in organizing your fostering and other volunteer activities.

CiviContribute would be my first choice in setting up adoption applications, complete with payment tracking and reporting.

Short answer, yes: it can be done with CiviCRM on a WordPress site. Might even find the BuddyPress plugin useful in tandem with Civi to energize social communication and networking among volunteers and related animal advocates.

Good luck Mark!