For chrome you will need to install chrome extension "Target VEC helper" which will inject target libraries. If you already have it, please go to Page delivery under setting menu and switch off "Enable enhanced composer". After this it should work.
Usually, I don't see any issues with Firefox, however you can try turning on/off "Enable enhanced composer"
@MilanCZ91 This issue maybe happening because google decided to change the default setting for the samesite attribute in cookies. you can find more info here: https://www.chromium.org/updates/same-site
As stated in the above link, the rollout is almost complete and thus VEC issues are expected.
Here is the recommendation to be able to create activities in Target :
1. Go to Go to URL : chrome://flags/#same-site-by-default-cookies in your chrome browser.
2. Disable the "SameSite by default cookies" flag
3. Relaunch the browser
Let me know if this helps?