I tried them after you emailed me about it before and was not impressed. There is no automated tool that can tell you definitively whether a site is accessible or meets accessibility laws or guidelines. Automated tools can be of some use in flagging potential things that need to be evaluated by a real person to determine if it is a problem or not. As such these kinds of tools tend to flag lots of false positives that can make you worry that your site has problems. The BOIA is a for profity company that wants to upsell you from their free reports to their paid reports and want to come across like they are some official authority with their name but they are not.
When I tried them the report had so many false positives it would be a real pain to find any useful report of a problem because of all the noise you have to filter out from the false positives. They reported my PayPal donate link as having a problem for example and they reported it thousands of times because my site has it on every page and I really don't think it does have any accessibilty problem. Their approach of wanting you to put their badge on your site showing how accessible your site is just a marketing ploy in my opinion trying to drive more people to their site so they can upsell them to their paid service. It seems like marketing based on fear. Fear that your site is not accessible generated by their scarey report, and I'm not keen on that kind of marketing approach. They don't have credibility with me and I certainly don't want to endorse them.
There are lots and lots of such tools that are probably better than theirs. The W3C Web Accessibility Initiative keeps a complete list of Accessibility testing tools and the BOIA is not on the list. I would suggest that if people do want to use automated testing tools for what usefulness they may offer they should try the ones on the W3C list.
But, no tool by itself can really definitively tell you whether a site has accessibility problems or not, as I said before they can be useful for pointing out things that a real person might want to evaluate but that is all.
Hope that helps,