Assuming you do not want to load any packages (you want a minimal change....) Also, you did not provide a minimal example, but the article class file contains
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
Because it is a TeX
\def, it can be overwritten with another
\def without the usual LaTeX existence crosschecks.
Here is an example assuming that you can typeset the UTF-8 characters. Otherwise, you'd have to replace them with LaTeX escaped characters.
Styczeń\or Luty\or Marzec\or Kwiecień\or Maj\or Czerwiec\or
Lipiec\or Sierpień\or Wrzesień\or Październik\or Listopad\or Grudzień\fi