You can do without Azure,
normally you will call the external web service inside a plugin (triggered when the contact is created/updated)
due to CRM Online limitations, the plugin must run in sandbox, for the call of the external webservice this means:
Only the HTTP and HTTPS protocols are allowed.
Access to localhost (loopback) is not permitted.
IP addresses cannot be used. You must use a named web address that requires DNS name resolution.
Anonymous authentication is supported and recommended. There is no provision for prompting the logged on user for credentials or saving those credentials.
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