I don't know how much flexibility gravity forms gives you. One approach that might be possible is to create the form in ASM, view source to get the action attribute from the form and the name attributes from all the form fields and use those with your gravity forms to make it send to ASM's post handler.
Hi Support Team, I am working with another shelter that has used the above method to successfully integrate Gravity Forms with Shelter Manager. While the use submission successfully makes it from our website and into ASM, we are trying to automate the list of adoptable animals that appears in the adoption form on our website such that it auto-refreshes and is always current with our list of adoptable animals in ASM. As an example, in ASM forms, the question "which animal are you interested in?" always shows animals currently available for adoption. We would like to have this auto-refreshing list available in our adoption form (via Gravity Forms) on our website. How does ASM generate the list of adoptable animals for its forms, and can we use similar code for the same effect in Gravity Forms?
SM injects the list into the page when it generates the form at the backend. You won't be able to do this. There is a service API call you can make to retrieve the list of adoptable animals as a JSON document, which you could use to populate a list. https://sheltermanager.com/repo/asm3_help/service.html#json-adoptable-animals-and-xml-adoptable-animals