The MssApiService kicks off a process that allows certain functionality to expose web pages. This service is named MssWebApiService-{customer name} and is installed as part of the MoversSuite upgrade process through the MssApiServiceInstaller.msi. The WebApiSerivce processes web requests for functionality such as the QuickBooks Integration.
Starting and stopping the MssApiService also starts and stops the WebApiService. You can also manage the process individually, i.e. restart just the WebApiService process through Microsoft Services.
Refer to the MssApiService topic for more information.