You have to provide arguments for the vector v and it also has to be in the form of a list of 3 components: as in:

Grad[Div[{v1[x1, x2, x3], v2[x1, x2, x3], v3[x1, x2, x3]}, {x1, x2,   x3}], {x1, x2, x3}]

which gives

$ \left\{\text{v1}^{(2,0,0)}(\text{x1},\text{x2},\text{x3})+\text{v2}^{(1,1,0)}(\text{x1},\text{x2},\text{x3})+\text{v3}^{(1,0,1)}(\text{x1},\text{x2}, \text{x3}),\text{v1}^{(1,1,0)}(\text{x1},\text{x2},\text{x3})+\text{v2}^{(0,2,0)}(\text{x1},\text{x2},\text{x3})+\text{v3}^{(0,1,1)}(\text{x1},\text{x2},\text{x3}),\text{v1}^{(1,0,1)}(\text{x1},\text{x2},\text{x3})+\text{v2}^{(0,1,1)}(\text{x1},\text{x2},\text{x3})+\text{v3}^{(0,0,2)}(\text{ x1},\text{x2},\text{x3})\right\} $