Didn't get a free LST

Did you try and go to STAP unit and click search above it, near the Lightspeed Tokens exclamation point?

It should then list you all the tokens , both free and paid.

I went to the LST-token overview, and selected the token from there.

I was forwarded to the shop, shown on the picture, where I can buy it for real money

Why exactly can't you access the webstore?

I must have messed up somehow, when I created the account a month ago, because I get a message, saying I'm banned for breaking the EULA.
I have created multiple tickets, asking why, but I can't get a clarification

I might not be the only one who was banned from the store:
https://forums.ea.com/discussions/swgoh-technical-issues-en/automated-ban-of-new-ea-accounts/12390501