This is similar to what I have experienced when trying to make appointments online. For weeks, the county health dept site would open at a set time and, for me, would simply freeze. Or it would say "server error." Five minutes later, it let me in, but all appointments were full. Who the heck was getting those appointments?
Then I tried pharmacies and grocery stores - and got results similar to yours.
However, when the county finally went (after a public outcry!) to an automated waitlist, we were contacted fairly quickly for husband's appointment. And once the over-60 rule went into effect, and I got on that waitlist, the automated invitation came through in a week or so.
There is a lesson to be learned. Don't make people compete for appointments. Just let them sign up, and either first-come-first-served, or oldest birthday first.