Internet communication can mean lots of different things today. We have webcams and photos which are often used and exchanged as a part of the “relationship” that people create via the internet. It is hard to know precisely what the questioner is asking here, but for the sake of simplicity, I will assume the question at hand involves only written communication.
As far as I know, adultery is interpreted to be a physical act in Halacha (Jewish law), though the law sets up parameters that guide us in making certain assumptions about impropriety. Traditionally, men and women who are not family and are not married are not permitted to be in a room alone together, as it is assumed that if they are, there was some form of inappropriate physical interaction. In other words, the boundaries we erect to guard against adultery reflect a sense that temptation is very difficult to resist. I would apply the same assumptions toward what mental health professionals call “emotional affairs.”
Dr. Gail Saltz, Associate Professor of Psychiatry at The New York Presbyterian Hospital Weill-Cornell School of Medicine, teaches that denial is a hallmark of emotional affairs. People convince themselves that since there is not sex involved, there is no betrayal of one’s spouse.
She writes, “Many people convince themselves that as long as there's no sex, it's not an affair. But it is. An affair really has to do with secrecy, deception of the partner and betrayal. It also has to do with the amount of emotional energy that you put into the other person and are no longer giving your partner. Most people are more disturbed by the breaking of trust than by the sex—it's what's most difficult to recover from when a partner has an affair." When we have a relationship with another person via the internet, we may invest the same type of emotional energy in it and be just as deceptive of our partners as we would be if that relationship were face to face. In other words, I would not distinguish between emotional affairs conducted through the internet or in person. They are both affairs. Your question is whether or not this violates the commitment one makes to his/her spouse.
In Genesis, the Torah teaches that Adam and Eve were created for one another. The text states, “for it is not good for a person to be alone.” We need intimate relationships in our lives in order to fulfill a deeply important need for companionship. The institution of marriage is meant to fill this need. In your question, you ask about the violation of one’s vows. In Judaism, we don’t get married through vows as one does in other religious traditions. We are married through the recitation of blessings, and one of them says, “Grant perfect joy to these loving companions, as You did for the first man and woman in the Garden of Eden.” We are re’im ha’ahuvim, loving companions. In my opinion, developing a relationship with another person that fulfills a deep need for emotional intimacy does violence to the commitment reflected in this blessing. You can’t be a loving companion to someone from whom you are hiding another relationship.
My recommendation is to use transparency as a measuring tool for appropriateness. If you are willing to share the dynamic of the relationship that you are developing through the internet with your partner, and s/he is comfortable with it, it seems that the relationship is appropriate. If you are hiding the frequency of your communications from your partner or the details of your relationship, it is an egregious violation of the trust you are responsible to create with your partner.