Does my legal husband who is going through the annullment process so we can wed in the Catholic Church authrorized to be the one to contact the ex spouse for all her information, or does the Father do that? If the ex spouse has already stated they did not want to participate, is he to keep probing until he gets the information needed, or can the Father work with out it?
Leave it to the Tribunal (or priest), unless they specifically ask your husband to speak with his ex-spouse.
All your husband need do is to supply the information requested by the tribunal - usually contact information if they have it. After that the trbunal will make the necessary contacts and follow-ups.
In my case I did contact my ex, through my son so that they would know what was going on and why. Sounds like your husband has already done this. If she does not wish to participate (not at all uncommon) she can simply ignore the the contacts by the tribunal.