Manufacturing Clean Energy and Energy Efficiency Act of 2021 This bill requires the Department of Energy (DOE) to establish a program within the Office of Energy Efficiency and Renewable Energy that provides grants to small- or medium-sized manufacturers for projects that will (1) improve their energy efficiency, material efficiency, cybersecurity, or productivity; or (2) reduce their waste production, greenhouse gas emissions, or nongreenhouse gas pollution. In order to be eligible for grants, manufacturers must have completed energy assessments from Industrial Assessment Centers funded by DOE or third-party assessors that provide equivalent assessments. Only projects that have been recommended in energy assessments are eligible for grants. The bill also authorizes Industrial Assessment Centers to provide technical assistance to small- and medium-sized manufacturers that have completed energy assessments to upgrade, retrofit, or replace systems or infrastructure.