Hello @Paul Lethin,
Thanks for reaching out to Microsoft Q&A.
Yes, it is possible for a non-admin user to create Azure free subscription under the tenant. Unfortunately, there is no control in portal to restrict subscription signups (other than license-based subscriptions).
Also, once the Azure subscription is created, any resources or services consumed within that subscription, including the free tier services, are associated with the tenant. The billing is usually associated with the Azure account and subscription, but the tenant that owns the subscription is responsible for all the activities including billing and resource usage.
Once user has created Azure free subscription you will receive USD 200 in credits which can be used within the first 30 days on most Azure services. Within 30 days of sign-up or upon exhaustion of the credits (whichever occurs first), you must upgrade to a pay-as-you-go subscription by removing the spending limit. This allows continued use of the Azure free account and select free services for the term. When you upgrade to pay-as-you-go subscription, it will be further considered as billable for that tenant.
Further, it will also give them access to stand up servers (like VMs) using that non-admin user account.
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Janaki Kota