I need to figure out if the assignment was sent for approval in Project Online, using REST or JSOM
I have not meet exact end-point which matches your requirement, but you can do a workaround with below;
JSOM
var value = PS.StatusAssignment.get_isConfirmed();
REST
GET http://<sitecollection>/<site>/_api/ProjectServer/EnterpriseResources('resourceid')/Assignments('assignmentid')/isConfirmed
