A quick Google gave me this overview of the first 9 documents you'll need ( and which Amt is responsible)
https://www.arag.de/auf-ins-leben/selbststaendigkeit/food-truck-genehmigungen-und-vorschriften/?cookieSetting=true
https://translate.google.de/translate?hl=&sl=auto&tl=en&u=https%3A%2F%2Fwww.arag.de%2Fauf-ins-leben%2Fselbststaendigkeit%2Ffood-truck-genehmigungen-und-vorschriften%2F
G-Translate makes it look like this, but you may have to G-translate yourself,
as clicking the link forces a cookie question
Of course in Hamburg, some are Bezirk (District) and some are for the State of Hamburg.
Name of the document
Where can I get the document?
Business license for restaurateurs
Business registration office
Tax registration form
Automatically from the tax office by post after registration of the trade
Health certificate
Health department
Health permit
Regulatory office
Food hygiene training according to EU regulation (EG)
First instruction at the health department
Instruction according to Section 43 Infection Protection Act (IfSG)
First instruction at the health department, follow-up instruction every 2 years by the employer
Proof of information for the catering trade
Chamber of Commerce and Industry
Immission approval
Trade inspection
Dispensing permit or permission
Commercial registration office of the local community