Hi Vikrant,
As I understand, you want to create and call custom azure web service in SharePoint online.
You could use provider hosted app for SharePoint online.
For more detailed information, you could refer to the articles below.
The article below is about connecting Provider Hosted Apps to SharePoint Online
https://my.theobald-software.com/index.php?/Knowledgebase/Article/View/133/0/how-to-create-a-provider-hosted-app-for-sharepoint-online-to-access-sap-data-via-erpconnect-services-core
The article below is about how to: Create a provider-hosted app for SharePoint Online to access SAP data via ERPConnect Services Core
https://auth0.com/docs/integrations/sharepoint-apps
The article below is about building your first Provider-Hosted App for SharePoint using Windows Azure Part 1
http://blogs.msdn.com/b/steve_fox/archive/2013/02/18/building-your-first-provider-hosted-app-for-sharepoint-using-windows-azure-part-1.aspx
The article below is about deploying SP2013 provider-hosted apps/Remote Event Receivers to Azure Websites (for Office 365 apps)
http://www.sharepointnutsandbolts.com/2013/07/deploying-sp2013-provider-hosted.html
Best regards,
Sara Fan