Erhvervsstyrelsen has fixed the issue on their end and participants can be created again.

NemHandel is currently experiencing technical issues with their test SMP which prevents new participants from being registered on both NemHandel and Peppol.
Erhvervsstyrelsen (Danish Business Authority) is looking into the issue but are unable to provide a date as for when this problem is solved.

We'll post an update here once there is an announcement from Erhvervsstyrelsen.

Existing participants can send and receive without issues.