Do You Need Witnesses to Take Shahadah?
Question

 
I am a young man who is married to a Muslim woman but her family is not Muslim. She became Muslim and learned some surahs and how to pray via the internet. After we got married I asked her: at whose hand did you become Muslim? She said: I became Muslim by myself. I said to her: You have to utter the Shahadatayn, so she uttered them before me. Is that correct?
Please note that we got married in the mosque before the Shaykh and a number of witnesses, and she got a certificate proving that she is Muslim.
Praise be to Allah.Can you take shahadah without witnesses?
In order for a person to become Muslim, it is not essential for him to declare his Islam before anyone. Islam is a matter that is between a person and his Lord, may He be blessed and exalted. 
If he asks people to bear witness to his Islam so that it may be documented among his personal documents, there is nothing wrong with that, but it should be done without making that a condition of his Islam being valid. 
This has been explained in more than one answer. Please see the answers to questions no. 655 , 6542 and 6703
Is marriage without a wali valid?
No marriage contract is valid without a wali (guardian) for the wife, but no non-Muslim can act as a wali for a Muslim woman. If she has no Muslim wali then the judge or the imam of the mosque or the local mufti (Islamic scholar) may act as her wali. 
In the answers to questions no. 7714  and 7989  you will find more details on the ruling on women in non-Muslim countries who do not have Muslim walis (guardians).
We ask Allah to bless you both and to join you together in goodness.
For more about the pillars of Islam, please see this answer: 13569
And Allah knows best.