What does a hornet nest foundress feed on?
I used the below articles to get the information.
1. the queen does collect both carbohydrates (nectar mostly) and protein (often bee flight muscles).


Only found one case study of one nest for this, so not sure. The queen did leave the nest during the period when it would be expected to have larvae. The articles I read did not mention food storage in the nest.
Same case study, the queen left the nest probably for longer than four weeks.


The developmental period was about 4I-42 days (egg 6 days, larva about 15 days, pupa about 20-21 days.
 from "Taxonomy, distribution and nesting biology of the Vespa bicolor group" (Hym., Vespinae) (1994) by Michael E Archer. found at https://www.academia.edu/4236740/Taxonomy_distribution_and_nesting_biology_of_the_Vespa_bicolor_group_Hym._Vespinae_1994_

This review article has a good overview. "Vespa velutina: a new invasive predator of honeybees in Europe" by Karine Monceau, Olivier Bonnard, and Denis Thiery.  DOI 10.1007/s10340-013-0537-3. accessed at https://www.doc-developpement-durable.org/file/Elevages/apiculture/maladies&amp;parasites/Vespa%20velutina_frelon%20asiatique/Vespa%20velutina%20a%20new%20invasive%20predator%20of%20honeybees%20in%20Europe.pdf

A case study of a captured nest (one queen, four workers) found that the queen stayed inside the nest permanently after 27 more days.

From "Observations on the colony activity of the Asian hornet Vespa velutina
Lepeletier 1836 (Hymenoptera: Vespidae: Vespinae) in France" Article in Annales- Societe Entomologique de France · January 2009 DOI: 10.1080/00379271.2009.10697595 downloaded from ResearchGate

There are also other articles that may have the answer, but in languages that I could not read.