Is the initial marriage between a Muslim man and a non-Muslim woman, followed by the wife's conversion to Islam, considered Islamically valid, or does it require renewal of the marriage contract?

A valid marriage that has fulfilled its conditions and pillars occurs only once. However, if the marriage was invalid (fasid), you must avoid the wife until her 'iddah (waiting period) from the invalid intercourse ends. Then, you may contract a valid marriage with her if you wish.

Some scholars are of the opinion that in the case of invalid intercourse that does not result in a child, the woman must observe 'istibra' (a short waiting period to ensure no pregnancy) before contracting a new marriage. However, if the intercourse could result in a child, then it is permissible to contract marriage with her without 'istibra', because in this case, the 'iddah is for preserving the lineage of the one who had intercourse.

Therefore, you must determine the status of your first marriage: If it was valid, then the marriage is sound as stated by the Imam of the mosque. If it was invalid and you are certain of the invalidity of the intercourse, then you must wait for the 'iddah to end.