The Imperial Germany Army issued an order 15 June 1898 stating that the flagbearers were to wear a gorget (breastplate) and a shield-shaped cloth insignia. This gorget tradition was continued not only by the Wehrmacht but also by several other organizations as can be seen below.

Reichsminister and Chef der Reichskanzlei SS-Brigadeführer Dr. Hans Lammers in 1937, note the gorgets worn by the SA men

(Courtesy of Bundesarchiv/Wikimedia, licensed under Creative Commons Attribution ShareAlike 3.0 Germany)

Deutsche Reichsbahn

Standard Bearer Gorget

Deutsches Rotes Kreuz (DRK)

Standard Bearer Gorget

Heer

Bahnhofswache Gorget
Feldgendarmerie Gorget (introduced 1938, also used by the Waffen-SS)
Feldjäger Gorget
Kommandantur Gorget (introduced 1937)
Standard Bearer Gorget (introduced 1936)
Zugwacht Gorget

Heer Railway Protection Gorget

(Courtesy of The Ruptured Duck)

Heer Standard Bearer's Gorget

(Courtesy of The Ruptured Duck)

Hitlerjugend (HJ)

Standard Bearer Gorget (introduced 1936)

Kriegsmarine

Marine-Küsten-Polizei Gorget

Luftwaffe

Feldgendarmerie Gorget
Regiment General Göring Gorget (introduced 1936)
Reichs-Luft-Aufsicht Gorget (two different versions existed)
Standard Bearer Gorget (introduced 1936)

NationalSozialistische Deutsche ArbeiterPartei (NSDAP)

Standard Bearer Gorget (issued 1939)
Streifendienst Gorget

Nationalsozialistische Kriegsopferversorgung (NSKOV)

Standard Bearer Gorget (introduced 1937)

Nationalsozialistischer Reichskriegerbund (NS-RKB)

Standard Bearer Gorget (two different versions existed)

Nationalsozialistisches Fliegerkorps (NSFK)

Standard Bearer Gorget (introduced 1937)

Nationalsozialistisches Kraftfahrkorps (NSKK)

Standard Bearer Gorget (introduced 1935)
Streifendienst Gorget
SA Flagbearer Gorget (used by the NSKK until 1935)
NSKK Verkehrs-Erziehungsdienst (introduced 1937)

NSKK Verkehrs-Erziehungsdienst Gorget

(Courtesy of Frans Baats)

Polizei

Motorisierte Gendarmerie Gorget
Standard Bearer Gorget

Reichsarbeitsdienst (RAD)

Standard Bearer Gorget
Streifendienst Gorget

Reichsluftschutzbund

RLB Gorget (two different versions existed)

SchutzStaffel (SS)

LSSAH Standard Bearer (introduced 1934 and used only by Leibstandarte SS Adolf Hitler)
SS-Funkschutz
SA Standard Bearer Gorget (used by the SS until 1938)
SS Standard Bearer Gorget (introduced 1938)
SS Streifendienst (introduced 1935)

Stahlhelm

Standard Bearer Gorget (two different versions existed)

SturmAbteilung (SA)

SA-Feldjägerkorps Gorget
SA Standard Bearer Gorget (introduced 1929, the first offical gorget with eagle and swastika)
SA-Standarte Feldherrnhalle Gorget (introduced 1935 and used by all members to the unit except the flagbearers)
SA Wache Gorget (possibly unofficial)

Technisches Nothilfe (TeNo)

Standard Bearer Gorget (introduced 1937)
Technisches Aufseher Gorget