Good day @danfortunato
This issue is not related to the Azure Virtual Machine directly. There is not Hotlink Protection service which is automatically added to virtual machine by default. Check that you did not added such extension to the VM. I never used it but there are some companies that provide such services. A Hotlink Protection service is an application which create another layer between the incoming requests and the web server (like a proxy in this behave).
It might be related to your site code/configuration (I develop such behavior in several web applications in the past and it is very common). Make sure that this is not related to specific site - create new blank site and test it.
It can also be related to configuration of the web server. If you search google for: Hotlink protection Apache, then you will found multiple posts on how to configure Hotlink protection.
And it can be related to the firewall configuration - I am speaking about your firewall in the machine and not the Azure firewall for the vNet.