Expediting Assistance to Renters and Landlords Act of 2021 This bill modifies the Department of the Treasury's Emergency Rental Assistance program by extending time limits on payments, allowing landlords to seek rents and arrears from the program without renter consent, and allowing renters to prove eligibility through attestation. The program provides grants to state and local governments to assist eligible households in paying rent and utility costs and other expenses related to housing incurred due to the COVID-19 pandemic. The bill extends the availability of program assistance from 18 to 24 months for eligible households. The bill also relaxes requirements for proof of eligibility by requiring only that households attest that they meet eligibility requirements instead of requiring proof of eligibility. It also allows for direct payment of funds to eligible households where the landlord refuses direct payment. Additionally, the bill allows a landlord to apply for program funds without the consent of a renter if the landlord agrees to certain terms, including a 120-day bar on evicting the renter for nonpayment. This bill also expands the stable housing services available under the program to additional low-income households. Further, the bill requires Treasury and state and local governments to increase outreach to renters and landlords and allows for coordination with public entities to provide application assistance services.