One year after a refugee's arrival into the US they are eligible to apply for their green card, or permanent residence, given they have completed all required immunizations and have received their USCIS packet from the Health Department. Refer to:

Their USCIS packet expires one year after it was issued. This document is useful for potential employers after the refugee's Work Authorization card has expired, and it also allows the refugee to travel outside of the country. 

In order to apply for a green card, refugees must go to a BIA accredited attorney. 

Elon Law's Clinical Law Center provides these services for no fee. However, the wait time for an appointment could be anywhere from 2 to 3 months. To contact them about an appointment their number is: 336-279-9299

Another resource is Franca Jalloh. She charges $120 per person for services and can be reached at 917-515-7556