Upload database records to AWS using TSQL?
perhaps you can share what exactly the software need, so without assuming things , I would describe approaches I use at my current work. 

I normally use powershell to extract the data from dbs, I use SMO [link] http://sqlmag.com/powershell/using-sql-server-management-objects-powershell

and then you can upload the output to an s3 bucket using write-s3object powershell command , this can be done from any customer environment which has access to internet, and with using the awspowershell module which you can download [link] https://aws.amazon.com/powershell/ and from a central EC2 in your aws you can read this data from s3-bucket , you can use read-s3object to get the data from the s3 bucket