You cannot.
The only way of renewing ID is doing it personally in your corresponding municipality in Poland. There is no legal possibility to delegate this to anyone else. As far as I know this contradicts EU guidelines, as EU countries have removed resident IDs for EU nationals, instead fully recognizing national IDs for all purposes. So there is still hope that in the future it might change. But right now the rules are clear. There is no way of doing that.
Polish Ministry of Interior has addressed the issue:
Obowiązujący stan prawny nie przewiduje możliwości złożenia wniosku o
wydanie dowodu osobistego oraz odbioru tego dokumentu za pośrednictwem
pełnomocnika. Zgodnie z rozporządzeniem Rady Ministrów z dnia 21
listopada 2000 roku (Dz. U. z 2000 r., nr 112, poz. 1182 z późn. zm.)
w sprawie wzoru dowodu osobistego oraz trybu postępowania w sprawach
wydawania dowodów osobistych, ich wymiany, zwrotu lub utraty, złożenie
wniosku o wydanie dowodu osobistego jak i odbiór dokumentu wymaga
osobistego stawiennictwa wnioskodawcy w organie gminy właściwym do
wydania tego dokumentu.
Rough translation
The current state law does not provide for any possibility to apply
for an identity card and pick up of said document by proxy. According
to decree by the Council of Ministers dated 21 November 2000 (Journal
of Laws of 2000 No. 112, item. 1182, as amended. Amended.) on the
model of the identity card and the mode of procedure for issuing
identity cards, their replacement, return or loss of, an application
for an identity card and the pick-up of the document requires
appearance in person of the applicant in the municipality competent to
issue the document.