I've never experienced this**, but I guess I've only done less than a dozen audaxes. Sounds like this was a preplanned thing. Did you get an email letting you know this was going to happen?
If it's something you want to do on a particular audax, you should contact the organiser. He/she may well be happy to allow this - especially if you have a good reason, as it's extra work for them. I wouldn't expect them to turn up early to give you your card, but they could allow you to collect receipts/atm slips around the route, effectively doing a DIY on the same day as the official ride. This would be necessary if you are a fast (>=30kph) rider, as you may get to the controls before they open.
However, if the time of an official event doesn't suit, I contact your
regional DIY organiser and just do it on a time and date that works for you.
**I haven't experienced a exactly what you described, but I signed up for The Willy Warmer in January a few years ago when truly terrible weather postponed it. All entrants could do it on their own any time in the next 3 weeks. Audaxes have to very flexible in winter.
Edit: most audaxes I have been on I could have set off before the official start time. Cards take a while to hand out, and so most people have them before the start time. This would be cheating, but I doubt anyone would call you on it. I mean, with audax, you are only really cheating yourself.