All five bodies were found in the family's home in Königs Wusterhausen outside Berlin on Saturday, Brandenburg state prosecutors said.
The 40-year-old father said in the note that he had secured a fake coronavirus vaccine pass for his wife.
But her boss spotted the fraud and planned to investigate, Cottbus prosecutors' spokesman Gernot Bantleon told AFP.
"The father expected that he and his wife would be jailed and their children taken away," Bantleon said.
Investigators believe the man killed the children aged four, eight and 10 and his 40-year-old wife before killing himself.
Neighbours called emergency services after seeing the lifeless bodies in the house.
Ongoing autopsies have so far revealed only fatal gunshot wounds to the head, prosecutors said.
German authorities have repeatedly warned of the spread of fake Covid-19 vaccination certificates, in a country hit especially hard by the latest coronavirus wave compared to its neighbours.
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