A short necrology of Henri Gustave Vogt can be found here:
https://gallica.bnf.fr/ark:/12148/bpt6k200265z/f5.item
His 1924 discourse on Henri Bazin here:
https://gallica.bnf.fr/ark:/12148/bpt6k200262t/f6.item
Some comments on his entry in the Académie de Stanislas in 1921 are here:
https://gallica.bnf.fr/ark:/12148/bpt6k2002594/f57.item.r=Vogt
He has received the Légion d'Honneur:
This says that he is born the 24th of January 1864 in Sermaize (Marne).
His "acte de naissance" says that his correct name is "Henry Gustave". He is the son of Jacques Georges Vogt and Charlotte Gabrielle Cavelier.
The last document in the Légion d'Honneur file contains a short biography!
Here comes a photography as a young student in 1881:
https://archive.org/details/ENS01_PHOD_1_1_30
New information:
I have got a scan of the front and back pages of his PhD thesis, where we learn that
The "Commission d'Examen" was composed of Hermite (président) and Appell and Poincaré (examinateurs)
The thesis is dedicated to Appell: "À Monsieur Appell, Hommage de respectueuse reconnaissance".
I would be tempted to deduce that Appell was his advisor.