Everyone is called to and allowed to join the Church, no matter what.
That being said: if they are married, they are married, period. The Church acknowledges their marriage as valid, even if it is civil. The issue comes forth because there is a previous marriage that has not been declared null.
Now, perhaps you mean he wishes that their marriage be convalidated by the Church as sacramental. Right now, neither of them is allowed to receive the Holy Eucharist until their marriage is convalidated by the Church.
So the steps would be: for him, to begin RCIA and talk to the priest about his situation; for them, to begin the request for the annulment of her previous marriage. Later, if the tribunal finds that the previous marriage was invalid, they will be able to request the convalidation.
Now: in the very unfortunate case that they end up in a divorce, then she would need 2 annulments in order to re-marry, and he would need 1 annulment if he becomes Catholic and wants to re-marry.
However, as long as they divorce and not re-marry, assuming they live in chastity, of coruse, it is my understanding that they can partake of the Holy Eucharist. But right now they cannot.