Step 1 would be to figure out how to write code to make a web request and save the response to a file, in a .Net framework language. This is not difficult once you realize T-SQL is not the answer.

Then if you need this functionality in SQL Server on demand, putting this in a CLR stored procedure would be the best solution. If it could be a scheduled batch process, then a script task in SSIS could do it. If this is a one-time data migration it could be a standalone program in any language, maybe even PowerShell.