There is no requirement that any one of the letters should come from people in academia. Also, if people submit recommendations directly to the admissions office, they are placed in your file. In other words, recommendations do not have to be submitted solely online.
It is unfortunate that there was a misunderstanding with the people administering the applications.
That said, I know they are doing a great job, almost perfect. Trying to resolve this differently, by contacting me, or the administrative assistant would have been more constructive than berating the staff and the program. I think the likelihood of the bigger loss being yours rather than the program's was somewhat higher by default. And then I read your post.