We need to create a website with shopping cart feature that receives login data from an external webservice and then at the end of the purchase process sends a JSON file with a set of data regarding the purchase details to an external webservice.
Can we use Toolset (maybe in combination with Woocomemrce) to build the website with the above features?
Toolset doesn't include any shopping cart feature itself, it relies on that of WooCommerce for e-commerce sites.
For what you are describing it will likely require quite a bit of custom coding, using some combination of the WordPress, WooCommerce, and/or Toolset APIs.