Integrating with a third-party SQL Server running locally

I have a little side-project where I’m faced with the task of integrating a cloud solution in azure with a database running locally on site. The database in question runs SQL Server 2008 and collects data in real-time from several cash desks on the site. The machine running the server has internet access but the network is not configured for accessing the server from the internet. I have complete access to the database locally (at least with SSMS).

What is the best way to sync the local data to the cloud? Mirroring? Some other way? Azure Data Sync? Since I’m designing a close to real-time system changes in the local database needs to be reflected as fast as possible.

I’m kind of a newbie when it comes to database management so any nudge in the right direction is much appreciated.