Biorthogonal multiwavelets are generated from refinable function vectors by using multiresolution analyses. To obtain a biorthogonal multiwavelet, we need to construct a pair of primal and dual masks, from which two refinable function vectors are obtained so that a multiresolution analysis is formed to derive a biorthogonal multiwavelet. It is well known that the order of vanishing moments of a biorthogonal multiwavelet is one of the most desirable properties of a biorthogonal multiwavelet in various applications. To design a biorthogonal multiwavelet with high order of vanishing moments, we have to design a pair of primal and dual masks with high order of sum rules. In this paper, we shall study an important family of primal masks --- Hermite interpolatory masks. A general way for constructing Hermite interpolatory masks with increasing order of sum rules is presented. Such family of Hermite interpolants from the Hermite interpolatory masks includes the piecewise Hermite cubics as a special case. In particular, a $C^3$ Hermite interpolant is constructed with support $[-3,3]$ and multiplicity $2$. Next, we shall present a construction by cosets (CBC) algorithm to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. By employing the CBC algorithm, several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a $C^1$ dual function vector of the well-known piecewise Hermite cubics is given.