There is no mention of any Swayamwara for Sita! Rama and Lakshmana were led to Mithilai by Vishwamithra to worship the divine bow of Shiva. Janaka explains its sanctitiy and informs his decision of marrying off his daughter Sita to whoever that lifts the bow and strings it. Many have failed to even lift the bow in the past. Rama, Lakshmana were shown the bow and Rama enquires if he can lift it. After Vishwamithra and Janaka assents, he simply lifts and breaks the bow. Janaka then wants to marry off his daughter to Rama, but Rama says he can marry Sita only after obtaining the permission of Dasaratha.
There is no reference of any Swayamwara or Rama and Sita meeting each other before marriage.