The first step is to find out if this is a real FHC, or something dreamed up by the local leaders. A real FHC has a separate unit number and shows up in the CDOL.
Family History Centers are actually run by a subsidiary of the church. It used to be called the "Genealogy Society of Utah" but I think they changed their name to "FamilySearch". Assuming this is a real FHC, you'll need to talk with the Family History Department. You can call CHQ and ask to be connected to them.
I should warn you that it's very difficult to create a FHC. They also have resources that you can't get from home. So I'd think about the closing very carefully. The Internet has not made them obsolete.
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