Is a marriage relationship built online recommended, and is there a way to tell Mother about the girl's engagement?

From a Shariah perspective, communication and correspondence between a man and a non-mahram (definition) woman are forbidden due to the potential for fitna (temptation, discord) and the attachment of the heart, which could lead to engaging in prohibited acts. The Permanent Committee for Scholarly Research and Ifta stated in their fatwas: "It is not permissible for you to correspond with a young man who is not a mahram to you, in what is known as 'acquaintance corner'; because that stirs fitna and leads to evil and corruption." The Prophet, peace be upon him, said: "(Indeed, this world is sweet and green, and Allah has made you vicegerents in it, and He will see how you act. So, beware of the world, and beware of women, for the first fitna of the Children of Israel was concerning women)" narrated by Muslim.

It is obligatory to repent to Allah and cease these correspondences, as the girl is a non-mahram to you and is not permissible for you. A happy marriage cannot be built upon disobedience. There is no harm in marrying her after inquiring about her religiosity, character, and the state of her family. If she is satisfactory, seek Allah's guidance (istikharah) and approach her guardian. Look for a suitable way to inform your mother about her, other than the internet.