(Disclaimer, work at Microsoft. Not on VS Code.) I expect you should be able to run Fiddler or Wireshark or similar traffic sniffers to see the requests.
At least in theory, it shouldn't be necessary to go to such an extreme, considering that VS Code is ostensibly FOSS and thus readily auditable for this sort of thing: https://github.com/Microsoft/vscode
This assumes, of course, that y'all aren't doing any weird code-injecting funny business when packaging it up for installation :)
I presume these requests are encrypted? So, it is hard to know what exactly is being sent.