Visa extension

is it possible for a Japanese company representative to extend their employee's sponsorship working visa whilst the employee is outside the country?

It seems like impossible. The main reason for that is the employee should be in Japan to extend his residence card and apply by himself the documents that will the company prepare for the residence extension.
However, if you mean that your residence card had already expired, while you were outside of Japan and could not return because of coronavirus, then you should contact with the Embassy of Japan in your country. I heard that those people can re-enter the country with the permission of Consul of Japanese Embassy. Double check that, maybe you have a chance to return with old residence card and renew that. Otherwise, you should apply for the CoE again.

Wish you good luck!

What are the exact circumstances?

Under normal circumstances, if the person with an expiring resident status has left Japan, and for some reason could not return to Japan before it expires, they would need to start all over, from zero, meaning starting with applying for a Certificate of Eligibility and then visa, with all the documentation required.

However, if the person has been living in Japan with a resident status, has left Japan with a re-entry permit, but is unable to return to Japan before the expiry due to COVID-19, the immigration authorities are allowing for an exceptional handling, or at least a simplified procedure, namely, the sponsor in Japan would still need to apply for a new CoE, but only with the application form and a statement of reason why you could not enter Japan, and they would process that in a shorter time than usual (they say).
This would be 3. (2) On the following page:
http://www.moj.go.jp/content/001323022.pdf

I do not know if this applies to your situation though. Best wishes.

Should be possible, but it would require sending the necessary documents to Japan (like residence card/passport) and that might not be practical. The applicant does not need to personally submit the renewal.