Yes, that will be a problem… but once you’ve gotten it (platform files and toolchain) downloaded, PlatformIO does cache them, so it won’t need to re-download them (unless you tell it to update the platform or framework files).
You could also try using the PIO Terminal (since you don’t have a project open - meaning it’s not available on the status bar - you can open it via the Command Palette - View -> Command Palette... and start typing terminal) - it has to be the PIO terminal - not one of the others - so that it recognises the platformio command.
… and installing the platform manually… then at least you’ll get a progress display…
i.e. for the Arduino AVR boards, you want
pio platform install atmelavr
or for ESP8266 boards you want
pio platform install espressif8266
This is the post I was thinking about re: twitter… but hopefully there is a better way to fix it… as Ivan seems to be suggesting system level blocking via the hosts file or blocking pioplus access to twitter.com via your firewall software?