Can't Register for a Developer Account
I have the same issue. The DNS name doesn't exist, so changing browsers is of no use.

Apple needs to either publish the domain name in DNS so that it resolves to an IP address or they need to update the HTML to use the correct name that does exist.

It appears that either Apple's DNS broke and they either haven't realised, don't care to fix it, or it is very difficult to fix and they haven't put a workaround in place (improbable IMFO).

Unless you know what the IP address should be and it's just a DNS issue, you could add a host entry as a workaround. However, if the IP address is unknown or if the server/infrastructure is down in addition to DNS then there is nothing we can do except wait for Apple and hope they fix it.