Short answer
Test it against WCAG, the guidelines the Justice Department points to. Run an automated scanner first, then use the site yourself with only a keyboard and with a screen reader. A scanner alone can't tell you: the W3C says such tools “can only assist” in judging accessibility.1. Run an automated scan
Free tools catch the mechanical problems quickly: missing image descriptions, low color contrast, form fields with no label. Good options are the axe DevTools browser extension, WAVE, and the accessibility section of Lighthouse built into Chrome. Run one on your homepage, a service page and your contact form.
2. Put the mouse away
Press Tab repeatedly from the top of a page. Can you reach every link, button and form field? Can you always see where you are? Can you open the menu, fill in the form and send it using only the keyboard? If you get stuck, so does everyone who can't use a mouse.
3. Listen to it
Turn on a screen reader: VoiceOver is built into Macs and iPhones, and NVDA is free for Windows. Listen to your homepage. Do images have sensible descriptions? Do buttons say what they do, rather than “button”? Does the page make sense read top to bottom?
4. Check the problems the Justice Department names
Its guidance lists poor color contrast, missing alt text, videos without captions, and mouse-only navigation as examples of barriers (US Department of Justice). If your site has any of those, start there.
Why a scanner isn't enough
The W3C, which writes WCAG, is clear that “tools cannot check all accessibility aspects automatically. Human judgement is required”, and that evaluation tools “can not determine accessibility, they can only assist in doing so” (W3C). A clean scan means the mechanical checks passed, not that real people can use the site.
Can a site be certified compliant?
There is no official ADA certification for websites, and anyone selling one is overstating it. What you can have is a dated record of testing against WCAG and the fixes you made, which is what we publish about our own site.
General information, not legal advice.