Firefox and Firefox Developer Edition both include an accessibility checker in their developer tools.
Does this tool catch everything that would cause a page to not be ADA compliant? We need to make sure that our web app satisfies the ADA's requirements.
No, no accessibility checker or combinations of accessibility checkers is effective at finding all accessibility issues.
You need someone who knows WCAG, possibly ATAG depending on the application and ADA compliance to test for you.
Accessibility checkers are tools to assists you, unfortunately we are a long way from an automated solution that catches everything.
You need experience and a deep understanding of the WCAG guidance, even if your goal is purely compliance (as I would encourage you to change mindset from "compliance" to "best user experience").
The reason we can't rely on tools yet is because it is difficult to automate some (most) things, for example logical focus order is not something that has been successfully automated yet. (Top tip - accessibility insights has a good tool for testing logical tab order, it still requires some manual work but better than just hitting tab alone.)
Or forms - yes tools can tell you if you are missing a label but they can't tell you if error messages are meaningful, if instructions are clear, if the form label is located close enough to the input to not cause confusion etc.
To give you an idea of how far away we are with automated tools, the UK Government's website accessibility team tested automated tools on the "worlds least accessible webpage".
Out of the 10 tools tested they found that 29% of barriers were completely missed by every tool. (So even if you ran all 10 tools you wouldn't pick up 29% of errors - including manual prompts to check things!)
The most effective at finding errors automatically was Tenon....it found 39% of the accessibility errors.
There is a break-down of how these accessibility testing tools performed here.
In short, they are tools that will help you find about half of the accessibility issues on your app / page.
For everything else you need someone who knows how to test your site and find accessibility errors.