If I use Froslass' "Spiteful Sigh" attack to put enough damage counters on Froslass to leave less than 40 HP on it, will its attack be prevented if Blizzard Town is in play?
No, its remaining HP is checked for Blizzard Town before you place the damage counters onto Froslass.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_38
SM11_38