If a caster wants to use a spell scroll as a reaction (casting a spell that has a reaction casting time), do they need to already be holding it in their hand? What if they have the War Caster feat?
You need to hold the scroll on your turn, and have it in your hand by the time you wish to read it.
Casting the spell by reading the scroll requires the spell's normal Casting Time.
This is required, because object interactions (like getting the scroll from somewhere) happen on your turn, and you need access to the scroll to read it, by the Spellcasting rules.
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind, or must have access to the spell in a magic item.
The Warcaster feat simply allows you to cast the scroll as a reaction, for an opportunity attack, if the spell meets the criteria.
The spell must have a casting time of 1 action and must target only that creature.
If the scroll's spell had a casting time of 1 Reaction, you don't need Warcaster to use it as a reaction. Further discussion here.