You will be creating a web api and hosting it in azure app service( api app).
Within the code for that given method you will the implementation within which you will make the database calls. Insert in table A and get the record ID from there and doing other CRUD operations.
Firstly if you are doing it via stored procedure or entity framework there are options to get the last inserted record id for e.g.
Get Record ID in Entity Framework after insert
Since all the db operations have to happen together or must be rolled back you should be doing it within transactions. From Entity framework 6 onwards you have this capability
https://msdn.microsoft.com/en-us/data/dn456843.aspx
You can also do the same in ado.net or having transactions within stored procedure.