The Capulets greatest moments of wonder may come during the fake death. It is rare for a child of Juliet's age to just die in her sleep. I think Capulet at least believes it is possible that her cause of death was MURDER:
Despised, distressed, hated, martyr'd, kill'd!
Uncomfortable time, why camest thou now
To murder, murder our solemnity?
If they suspected anything else, they might have suspected suicide because that could be a cause for a death at a young age. This all occured in Act IV, scene iv.
The friar was very quick to encourage the family that Juliet's death is a blessing. This caused them to get out to the church right away and turn the planned wedding into a funeral.
Due to Juliet's outright refusal in getting married in Act III, Scene 5, where she responds negatively to her mother's insistence in marrying Paris, it would not be surprising if the Capulets assume Juliet may have killed herself to avoid getting married. However, it is not written anywhere in the play.
In addition, Juliet tells her mother that she would rather marry Romeo than Paris in Act III, Scene 5. Therefore, Lady Capulet could make a connection between Juliet's death and Romeo. This could implicate Romeo as the cause of Juliet's sudden death. There are a few possible choices; it is up to the reader to decide what may be the logical choice.