The woman chose to marry invalidly and then live as though she were a validly married woman—this is a grave situation. She will not be allowed to receive Communion as long as she ignores this. I can only recommend that she repent and go to confession, and that they commit to living as brother and sister until the situation is rectified (i.e., annulment and convalidation). If she does this, she can receive Communion.
The Catechism of the Catholic Church explains,
Today there are numerous Catholics in many countries who have recourse to civil divorce and contract new civil unions. In fidelity to the words of Jesus Christ—”Whoever divorces his wife and marries another, commits adultery against her; and if she divorces her husband and marries another, she commits adultery” (Mark 10:11-12) —the Church maintains that a new union cannot be recognized as valid if the first marriage was. If the divorced are remarried civilly, they find themselves in a situation that objectively contravenes God’s law. Consequently, they cannot receive Eucharistic Communion as long as this situation persists. For the same reason, they cannot exercise certain ecclesial responsibilities. Reconciliation through the sacrament of penance can be granted only to those who have repented for having violated the sign of the covenant and of fidelity to Christ, and who are committed to living in complete continence. (CCC 1650)