How do I code a command block so that it checks for if a player is wearing a specific Helmet called "☀ Disco Helm ☀"?

here is the working command. For an explaination, see below.

/execute as @s[nbt={Inventory:[{Slot:103b,id:"minecraft:leather_helmet",tag:{display:{Name:"{"name":"☀ ","color":"gold","italic":"false"},"name":"D","color":"dark_red"},{"name":"i","color":"red"},{"name":"s","color":"gold"},{"name":"c","color":"yellow"},{"name":"o ","color":"green"},{"name":"H","color":"aqua"},{"name":"e","color":"blue"},{"name":"l","color":"light_purple"},{"name":"m","color":"purple"},{"name":" ☀","color":"yellow"}"}}}]}] run setblock 1 4 4 minecraft:redstone_block replace`

You need to specify the color for each change in color. That was the only error in your command. I have to admit, it is painstaking. If the above command doesn't work, please tell me what color you set the spaces to in the comments. (Yes, spaces have a color)

You're welcome, ShadeOfLight

I see a few problems, and I have one suggestion.

In order to spot the problems, let's expand your NBT into a layered structure:

0: nbt={
1:     Inventory:[
2:         {
3:             Slot:102b,
4:             tag:{
5:                 Name:"{"☀ Disco Helm ☀"}"
6:             }
7:         }
8:     ]
9: }

The problems:

1. The item structure is different—the Name tag goes inside another tag named display.
   Wrong way:

   tag:{
       Name:'...'
   }
   

   Right way:

   tag:{
       display:{
           Name:'...'
       }
   }
   

2. The raw text in the Name tag is malformed. When checking if the name matches, the system does a plain text check on the raw JSON source, and does not look at the final name after rendering. So hold the item in your main hand, and run the following command in your chat:

   /data get entity @s SelectedItem.tag.display.Name
   

   Look for the text in GREEN, that represents the raw JSON source of the name. In the original command, place the GREEN text into the Name tag. Make sure you escape double quotes where necessary.