Does a caster need a free hand to cast a spell from a spell scroll? What if they have the War Caster feat?
Normally, yes, because they have to unroll the scroll.
To cast a spell from a scroll, you must read it. A scroll is usually stored rolled up to protect it from damage, so you would first need to unroll it (after taking it out of the scroll case if it's in one). Normally this would require using your hand, though there are alternatives, such as having someone else hold the scroll in front of you, or doing something very silly.
If the scroll were already, say, unrolled on a desk, then you could simply walk up and read it, but that's not typically how scrolls are used.
At the table, I assume that for spells that have a somatic component you'd use the hand that's holding the scroll. This is mostly just for simplicity, but it's somewhat justified by the fact that you can do that with a spellcasting focus, and a scroll is like a focus in that it acts as a substitute for material components.
Short answer: yes.
The description for a Spell Scroll includes the following line:
If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components.
Note that it specifically mentions material components but does not mention verbal or somatic components.
The entry on somatic components includes:
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
Since the spell scroll does not remove the somatic component of a given spell, you would still be required to perform the required motions with a free hand, per the above.
There is a possible exception, though: if the spell on the scroll does not require a somatic component, you would not need to have a free hand.
The War Caster feat mentions:
You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.
This seems straightforward, though a spell scroll is neither a weapon nor a shield.
This DM might allow someone with the War Caster feat to use the spell scroll without the somatic component, if the spell required one. I can't think of any particular scenarios where such a minor breach of action economy would "break the game" per se.
Regarding magic items
The rules for magic items include a section on spells. This section includes (emphasis mine):
Some magic items allow the user to cast a spell from the item. The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise.... Certain items make exceptions to these rules, changing the casting time, duration, or other parts of a spell.
But we also have another rule in Specific Beats General (emphasis mine):
...many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins.
In this case, the rules for the Spell Scroll are more specific and cover almost all the same details as the general magic item spellcasting rule, so they should take priority, including the part about material requirements.