Is it possible for non-Skokie residents to obtain a library card?

Yes.. From the library website: SUBURBAN RESIDENTS If you live in another town that is part of the RAILS public library system, bring your local library card and a photo ID that shows your name and current address in order to register that card with us. Nonresidents can check out up to 30 items. Borrowing limitsdo apply. If you live in Chicago you can buy a Skokie Public Library card for $350 for 12 months or $175 for 6 months. For staff at reciprocal libraries Please use this form to verify the current standing of Skokie Public Library patrons who would like to register their cards at your library

for free I don't think

Yes, I have Chicago friends who have purchased a "membership" for Skokie Library.