Since today, the Azure portal offers to send invoices directly to an email address instead of downloading your invoice every month.
See the announcement at Now available: Azure invoices emailed direct to your inbox.
The new menu "Send my invoice" is available in the "Billing" section. Click the "Opt In" button and configure the email address you want to use for getting the monthly invoices.
Important: You must be the account administrator to configure this setting.
This is the good message. Unfortunately, this service only is available for certain Azure subscription types and for example not for CSP subscriptions. Anyway, see, if this option is available in your subscription to automate getting invoices. For all other automation use cases, there is the Azure Billing API.