No, unfortunately meeting recording is not supported with the Family edition. You can refer to this article for detailed comparison of the features available across the various editions: https://support.microsoft.com/en-us/office/learn-more-about-subscriptions-for-microsoft-teams-free-1061bbd0-6d97-46a6-8ca0-21059be3eee3
If meeting recording is a required feature, you must use one of the "commercial" plans.