Is it valid to conclude a new marriage contract via Skype, with the presence of a guardian and two unified witnesses, for a wife living in the West who has no guardian, given that the husband is a quadriplegic?

It is not possible to judge your wife's father's actions as major shirk (polytheism) without knowing them. If they were like invoking other than Allah or sacrificing to other than Him, and he did not repent from them after the establishment of proof, then he exits Islam and the marriage becomes invalid, because the guardianship of a disbeliever over a Muslim woman is invalid by consensus.

If the father has repented, then he has the first right to renew the marriage contract. Otherwise, the guardianship transfers to the closest Muslim male relative (asabah). If there are no Muslim male relatives, the woman is permitted to appoint a just Muslim man to marry her. Appointing a guardian via modern communication methods is permissible if manipulation is safeguarded against.