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Thanks for the pointer from FrédéricC, it appears that Sage expects that the definition of polyhedron over ZZ is strict with all vertices being integral upfront. The behavior I expected can be achieved by first defining a polyhedron over QQ, computing its integral hull, and then changing base ring to ZZ like

P = Polyhedron( ieqs=[(-1, 1, -1, 0, -4), (2, -1, 1, 0, 4), (0, 1, 1, 2, -2), (2, -1, -1, -2, 2), (0, 0, -1, 1, 1), (1, 0, 1, -1, -1), (0, 1, 0, 0, 1), (1, -1, 0, 0, -1)], base_ring=QQ, backend='normaliz' )
Q = P.integral_hull().change_ring(ZZ)

Thanks for to the pointer from FrédéricC, it appears that Sage expects that the definition of polyhedron over ZZ is strict with all vertices being integral upfront. The behavior I expected can be achieved by first defining a polyhedron over QQ, computing its integral hull, and then changing base ring to ZZ like

P = Polyhedron( ieqs=[(-1, 1, -1, 0, -4), (2, -1, 1, 0, 4), (0, 1, 1, 2, -2), (2, -1, -1, -2, 2), (0, 0, -1, 1, 1), (1, 0, 1, -1, -1), (0, 1, 0, 0, 1), (1, -1, 0, 0, -1)], base_ring=QQ, backend='normaliz' )
Q = P.integral_hull().change_ring(ZZ)