When I play the game and update the board in game, it doesn't show it on trello. Why?

Your CardName variable records the value of the Text property when the script is first run; it doesn't change when (what I assume is) the TextBox changes. This can be remedied by changing the CardName variable to reference the object itself, and using CardName.Text in the rest of the script:

local ap = require(game.Workspace.TrelloAPI) --// Usage of local variables is recommended.
local BoardID = ap:GetBoardID("DCCC Voting Board")
local ListID = ap:GetListID("Voting On", BoardID)
local CardName = workspace.TextBoard.SurfaceGui.MainText --// Reference the object.
local CardID = ap:GetCardID(CardName, BoardID)
---------

script.Parent.MouseButton1Down:connect(function()
    CardID = ap:AddCard(CardName.Text, "Voting is in the Comments", ListID)
    local hint = Instance.new("Hint", workspace)
    hint.Text = "Trello Card Created for Bill: " .. CardName.Text --// Index the Text property.
    wait(3)
    hint:Destroy()
end)

Hope this helped.