Best way to establish local database online for payment application integration

A client I am working for needs their data accessible online. I need to be able to from my own API and application service running on the web write PUTS and do GETS from the existing database. They are running an SQL Server, so I am thinking just making this accessible online.

would it be best to run a complete Azure copy SQL relational database? they are currently running Swiftpos. or should I run a CosmosDB database?

For starting out I am thinking that an Azure Cloud database would be best purely for handling traffic. whenever a product gets updated in the database I would need to capture this and push it to the cloud database.