In the meantime, there are a number of sites who offer online event entry services - the one that my own (orienteering) club uses is one called Fabian4, who also do entries for various other sports.
Most of this stems from the fact that Audax events are put on by the individual organisers, not by Audax UK.
It's not because a good system doesn't exist; Audax UK offers an online entry system on their website to any organiser who wants to accept entries that way (and get payments via paypal). As IanH says, not all organisers want to do this, many are 60+ and don't even own a computer. Online entry is on the increase as younger and more computer savvy organisers are taking over.
It would take a significant reorganisation of the club for it to be able to do everything centrally and online, plus it would mean a non-insignificant increase in fees as you move from lots of people keeping under the limits for 'non-business' users to having to pay for commercial services.
It ain't broke so why fix it. Audaxes run perfectly well with the numbers they attract, many simply wouldn't work at all if they attracted the hundreds of people that sportives do. I like the fact that I can do them and see many familiar faces, that's definitely part of what an Audax means to me.