  In this paper, we prove determinant formulas for the $K$-theory classes of
the structure sheaves of degeneracy loci classes associated to vexillary
permutations in type $A$. As a consequence we obtain determinant formulas for
Lascoux-Sch\"utzenberger's double Grothendieck polynomials associated to
vexillary permutations. Furthermore, we generalize the determinant formula to
algebraic cobordism.
