I have a Scoreboard dummy objective called 'State', and I have an Armor Stand tagged with 'Game'. How do I make a fill clock that runs when that tagged Armor Stand has a state of '0'?

This is my current command:

-- top command block (impulse, needs redstone)
execute as @e[tag=Game, limit=1] if score @s State matches 0 run fill ~ ~-1 ~ ~-20 ~-1 ~ minecraft:redstone_block

-- redstone block in between 

-- bottom command block (impulse, needs redstone)
execute as @e[tag=Game, limit=1] if score @s State matches 0 run fill ~ ~-1 ~ ~-20 ~-1 ~ minecraft:stone

There are two things that need to be fixed:

First, change the second command to:

execute as @e[tag=Game, limit=1] if score @s State matches 0 run fill ~ ~1 ~ ~-20 ~1 ~ minecraft:air

The second problem is that your Armor Stand has, by default, no score (no score is different then '0'). To fix this, use:

/scoreboard players set @e[tag=Game] State 0

This should work.