Is SQL Server Mirroring dependent on SQL Server Agent running?
Mirroring is not dependent on SQL Server Agent running.  Communication between primary and secondary happens using TCPIP ports.  In database mirroring, principal and mirror server communicate through TCP connection and write data change directly to remote transaction log file. 


Database Mirroring Overview
COMPARISON BETWEEN MIRRORING VS LOG SHIPPING AND MIRRORING VS
CLUSTERING


But if you want to monitor and get alerted you need the agent job running which get created automatically when you set up mirroring. Details below:


Monitoring Database Mirroring (SQL Server)