CDC recommends that refrigerator and freezer temperature logs be kept for at least 3 years. See page 10 of the Vaccine Storage and Handling Toolkit, available at www.cdc.gov/vaccines/hcp/admin/storage/toolkit/storage-handling-toolkit.pdf. The reasoning is that it is useful to be able to look back at the record to help determine if a unit is developing a problem.
Individual state Vaccines For Children (VFC) programs may have additional requirements for retaining temperature logs. You should contact your state program for this information. Contact information for state immunization programs is available at www.immunize.org/coordinators.