When ESXi host where vCenter VM is placed goes down, as HA is enabled, vCenter VM will get restarted on other available host (or dedicated failover host) in the HA enabled cluster. vCenter is required to just manage & configure vSphere HA. Once configured, even if vCenter goes down, HA will work as HA agents are there on each ESXi host. Note that failed over vCenter VM will restart, hence there will be downtime till the VM gets restarted and all the vCenter server services also gets restarted.
In case of HA failure, VMs on failed host do not vMotion. It gets unregistered from failed host >> VMs will be registered on available host & restarted. in case of vMotion there is no downtime but in case of HA failure there will be minimal downtime as VMs will be restarted.