I had the same issue for about a week or so. I tried contacting apple support through the "email" option in the contact us page but there was no response from them. Then after a while I was able to request a call back from apple and only then did they ask me more details for verifying my identity and organization. I submitted the documents but then was again in the endless waiting loop. Then after a few days I had to request a call back from them again, told them I submitted all the documents and that's when I was able to proceed to making the payment for the program.
You can find the phone/call option here
Contact us. Although this call button keeps appearing and disappearing at very random times. So i'd suggest you to keep looking at it from time to time.
Topic:
Developer Tools & Services
SubTopic:
Apple Developer Program
Tags: