Is it possible to carry out multiplication of BGW protocol without communication?
No, it is impossible. We do not know of any way to perform this randomization step without interaction; furthermore, it was shown in this paper that multiplication require interactions in secure computation, in a variety of important settings (including all forms of BGW-style secure computation). Share conversions work for some restricted type of functionalities, but do not allow to evaluate generic multiplication gate over $\mathbb{F}_2$ on secret shares without interaction.

Note, however, that we can have multiparty computation with no interaction (except for sending an encryption of the input and sending back an encryption of the output) in the computational setting, using fully homomorphic encryption (which we can build assuming the LWE assumption).