Does Amazon Web Services (AWS) RDS support locally redundant storage (LRS) for Microsoft SQL databases?
Amazon doesn't call it LRS, but the Elastic Block Store (EBS) storage which backs RDS (see RDS FAQ, "What is the hardware configuration...") is automatically replicated across multiple servers (see EBS product page) within the same Availability Zone (i.e. locally). 

Data resilience is extremely high for RDS databases.