@CY ,
Thanks for posting this question.
For a deeper investigation into this we would need your subscription ID and Azure webapp details, which are PII, so I’ll follow-up with you privately.
For purchasing App Service Domain name, remove the spending limit on your subscription.
Also, you cannot buy App Service domains with free subscription credits. As mentioned in this doc section -Prerequisites
As for creating a support ticket without a support plan/for Quota request-- With Basic plan, billing and subscription management support’ is available to all the customers.
Please see, this – Support Plans comparison and FAQs
Note: Kindly do not share any PII data on the public forum/post.