Vorgang
process
A sequence of connected events or actions; a process or procedure.
Der gesamte Vorgang von der Bestellung bis zur Lieferung dauerte nur zwei Tage.
The entire process from ordering to delivery took only two days.

process / occurrence
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process
A sequence of connected events or actions; a process or procedure.
Der gesamte Vorgang von der Bestellung bis zur Lieferung dauerte nur zwei Tage.
The entire process from ordering to delivery took only two days.

'Vorgang' (process/occurrence) consists of the prefix 'vor' (before/in front of) and the noun 'Gang' (course/walk, from 'gehen' — to go). It describes something that takes place or 'vor sich geht' (goes on/happens).
Die Fotosynthese ist ein komplexer biologischer Vorgang.
Photosynthesis is a complex biological process.
You can recognize the meaning of 'Vorgang' (process/occurrence) from the context. In a technical or scientific context, it means 'process'. In everyday language or with the police, it describes an 'occurrence'. In an office or at a government authority, it means a 'file' or 'case file'.
Die Gärung von Wein ist ein chemischer Vorgang.
The fermentation of wine is a chemical process.
In the meaning of 'Prozess' (process) or 'Ablauf' (procedure/course), 'Vorgang' (process) is often combined with adjectives such as 'automatisch' (automatic), 'komplex' (complex), 'natürlich' (natural), or 'chemisch' (chemical).
Das Speichern der Daten ist ein automatischer Vorgang.
Saving the data is an automatic process.
The meaning 'Akte' (file) or 'Fall' (case) is typical of office and administrative language (official German). In this context, a Vorgang is often 'angelegt' (opened/created), 'bearbeitet' (processed), or 'abgeschlossen' (closed/completed).
Die Sachbearbeiterin hat für Ihre Beschwerde einen neuen Vorgang angelegt.
The case officer has opened a new case file for your complaint.
case / fall
'der Fall' and 'der Vorgang' can both refer to a particular event or matter that one is looking at. 'der Fall' emphasizes the concrete situation or case, for example with a problem, an investigation, or in court. 'der Vorgang' emphasizes the process more: several steps that belong together. In administrative settings, 'Vorgang' can also mean the case file for a matter. The words are easy to confuse because people often talk about both in offices or government offices. If the situation itself is important, 'Fall' fits; if the sequence of events or the handling of the matter is important, 'Vorgang' fits better.

Der Vorgang wird im Bürgeramt noch geprüft.
The case file is still being reviewed at the citizens' office.
Falls ihr morgen Zeit habt, können wir den Fall gemeinsam besprechen.
If you have time tomorrow, we can discuss the case together.