My cousin was supposed to fly back to Poland on May 27, but we just found out yesterday that her flight is cancelled and the earliest flight back is mid July - her six-month travel visa will expire June 12. We must extend her travel visa - we have no choice - how can we do this and not have to pay any fees... this all is due to the coronavirus.
If you haven't already done so, contact your embassy for advice on how to return to your home country. Embassies are organizing repatriation flights that are not advertised and only available by contacting the embassy. For example, there was a Poland-to-US repatriation flight on April 24. Although there may not be another repatriation flight available immediately, it is likely that one will be organized in the next 6 weeks between now and June 12. It may also be possible to return to your home country via a third country (eg return from US to Poland via Germany), but this would be safest if done with advice from your embassy.
Your only real options lie within the guidelines set out by USCIS here. Since your cousin needs a visa, she must apply for an extension. No way around it.
In summary, they state the following:
Generally, nonimmigrants must depart the United States before their authorized period of admission expires. However, we recognize that nonimmigrants may unexpectedly remain in the United States beyond their authorized period of stay due to COVID-19. Should this occur, the following options are available for nonimmigrants:
Apply for an Extension. Most nonimmigrants can mitigate the immigration consequences of COVID-19 by timely filing an application for extension of stay (EOS) or change in status (COS). U.S. Citizenship and Immigration Services continues to accept and process applications and petitions, and many of our forms are available for online filing.
Unfortunately that costs money, but it seems it's the only real mechanism the US has for extensions, even in times of a pandemic. USCIS do offer a fee waiver application here