Jotform offers the ability to easily add forms to a website, irrespective of the platform used (WordPress, Wix, etc).
To get started, you have two options (1) build from scratch, or (2) clone and edit an existing form.
Build from scratch
One of the reasons I recommend JotForm is the support community is outstanding. For all the information you could possibly need on getting started, visit their website.
Clone (copy) and edit an existing form.
For more details, go to their web page:
Below are links to sample forms, which you can clone for your animal rescue: