The OpenApply API has 2 versions of its public API - v1 and v3.
In the v3 API you will be able to retrieve students custom fields both from a generic "students" search, and by entering the student's ID. In the v1 API the generic "students" endpoint does not return custom fields, only a specific student search will retrieve custom fields.
The first type of call returns all students, but only returns their global (standard) fields.
The second type of call returns an individual student (the number is the OpenApply ID of the student), and will return all of their fields (including custom fields).
The endpoints are organised this way in order to prevent very large data requests on the system. However, you can, for example retrieve all students to find out their OpenApply IDs, and then retrieve an individual student, to get their custom fields.
When fetching all students, you can also use parameters like since_update, or since_id to identify which students have been updated recently, and fetch just those students that have been newly added, for example.