Description
Contrary to what the name suggests,
ToricVectorBundle may well encode a toric reflexive sheaf that is not locally free.
cartierInd computes the Cartier index, that is, the smallest nonnegative integer i such that the pullback of the bundle under the ith toric Frobenius becomes locally free. In case of the reflexive sheaf of a Weil divisor D, this is the smallest i such that iD is Cartier. This method works well only on simplicial toric varieties.
cartierInd calls internally the method
toricChernCharacter.
i1 : P112 = ccRefinement transpose matrix {{1,0},{0,1},{1,2}};

i2 : D = {1,0,0};

i3 : L = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( D, i> matrix{{i}}));

i4 : details L
o4 = HashTable{ 1  => ( 1 ,  1 )}
 2 
 0  => ( 1 , 0)
 1 
 1  => ( 1 , 0)
 0 
o4 : HashTable

i5 : cI = cartierInd L
o5 = 2

i6 : isLocallyFree L
o6 = false

i7 : L2 = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( cI*D, i> matrix{{i}}));

i8 : isLocallyFree L2
o8 = true

Caveat
This method works for a toric reflexive sheaf, which is locally Weil (see
isLocallyWeil), on a toric variety, whose fan is covered by cones of maximal dimension.