If you cannot login to Unicount with your Estonian e-resident card or resident ID-card then most probably your computer is missing the latest version of Estonian ID-card software called DigiDoc or your browser does not have the necessary plugins installed and enabled.
Some possible solutions:
Check whether your card works on the government site eesti.ee
If they do please install the latest DigiDoc software to your computer
Make sure that you have browser plugin installed and check that it is enabled
Then restart your computer
You can also follow this step-by-step guide for accessing web based services such as Unicount. Below you will also find links to goverment made video manuals for the most common operating systems and browsers.
Digidoc software installation manuals
Windows operating system
MacOS operating system
Linux operating system
Browser plugin installation manuals
If still no luck we would suggest trying to use your card to activate Smart-ID signature app on your mobile device. It has the same legal status as ID-card in Estonia and with it you can log in and sign without card reader on any smart device.