As atleast two people must enter keys into assigned lock,so the condition for vault to open is:-
k1k2 + k2k3 + k1k3
3 nand gates for there complement
(k1k2)' and 2 others.
((k1k2)'.(k2k3)')' 4th gate
(4th gate.(k1k3)')' 5 th gate
(4thgate.5thgate)' 6th Nand gate which gives k1k2+k2k3+k1k3