The API Gateway is packaged in an OVA image which can be used by vMotion, however the API Gateway application itself is a very time sensitive application and thus isn't always a great candidate for migration to different servers by use of vMotion.
While vMotion often works fine on the API Gateway, certain conditions may cause unintended consequences such as slow performance, freezing, or being inaccessible for a period of time. The conditions that generally cause this is a high load on the Gateway server traffic or processing wise, and if the vMotion task doesn't complete in a reasonable amount of time.
In such a case where the API Gateway suffers from some unintended consequences, the best way to resolve is to simply reboot the VM. Sometimes rebooting only the Gateway service is not enough and the whole VM should be rebooted from the VMware ESXi admin control panel.
If possible, use vMotion only when traffic on the API Gateway is low and closely monitor it during the process. If vMotion runs frequently automatically (often due to an over-utilized ESXi cluster), it may be wise to disable vMotion on the Gateway VMs to avoid any performance issues.