Usage of API

[deeprobin]

What are the terms of using this API? Am I allowed to distribute the data to customers?

[tafaust]

IMHO it's the same terms as if you'd use the Handelsregister directly.
This is not legal advice. Nor do I know your business neither your customers.
Seek legal advice on this issue.

[deeprobin]

Upon request, I was forwarded to the terms of use (Nutzungsbedingungen), which do not really help me with regard to the dissemination of data.

[tafaust]

I'd consult a lawyer to be really sure. The Handelsregister can only let you know about their ToS (in conformance with their issue to take actual responsibility for the service/data they provide).
Using that data to your liking in accordance with EU / local law is your business. I'd make sure to comply with GDPR because it is the most strict data protection law I've seen so far.

Again, this is not legal advice.
Seek real legal advice from a real lawyer.