No. Virginia is an open primary state which means that any qualified voter can vote in either party's primary election. Virginia does not have party registration in its voter registration process. In other words, citizens do not designate a political party affiliation when registering to vote in Virginia. The only restriction is when more than one political primary is held on the same day, known as a dual primary.
While Virginia election law stipulates that any qualified voter may vote in either political party's primary, no voter may vote in more than one political party's primary on the same Election Day.