Can I purchase a round trip ticket to Chicago from Los Angeles using my green card for identification?
ID requirements for domestic flights are set by TSA and can be found at https://www.tsa.gov/travel/security-screening/identification.
As you can see, permanent resident cards (aka green cards) are accepted.
If you have a compliant driver's license or state ID (or another acceptable form of identification), you don't even need to show your green card.
You should carry the green card with you because of the law that requires you to keep it in your personal possession at all times, but there is no requirement to have any particular travel document when traveling domestically in the US, and you can't be prevented from traveling without your green card (unless you are arrested for not having it, which is not something that happens frequently as far as I am aware).