The question is not whether the tower can operate. Class D towers exist and operate regardless of the weather -official at the site or otherwise. The tower will still be operating to clear IFR aircraft if the airfield is IMC or VFR aircraft requesting a SVFR clearance through the Delta. In most cases the wx will be generated right at the tower. Whichever source the tower is using to obtain the weather ( Controller observations, contracted observer, or other source.) The weather will always be "official." The class D technically exists when the wx is VFR, so an aircraft operating VFR in VMC conditions must request permission to enter/transit from the tower. But in IMC the tower is still there and the aircraft still cannot enter what would constitute the Delta area without permission. BUT that permission may be transparent to the pilot - the Approach controller working your aircraft may coordinate your passage with the tower and you'll never be aware of it.
