How do I make this gui open with a key instead of clicking the button?

Well, in your script, you opened the GUI VIA MouseButton1Down event. If you want to open it with key, you would want to use the InputBegan event of the GameService 'UserInputService'.

local frame = script.Parent 
local close = frame:WaitForChild("Close") 
local shop = frame.Parent:WaitForChild("Shop")
local uis = game:GetService'UserInputService' -- Define it
local opened = false -- Tell the game that they haven't opened yet

uis.InputBegan:Connect(function(key, gameProc) -- Connect to the function once a button press is detected
    -- Key is the key that is pressed and gameProc basically returns a boolean whether the key was pressed in chatbox or not.
    if key.KeyCode == Enum.KeyCode.L and not gameProc then -- If the key == "L" and they're not chatting
        if not opened then -- If opened is false
            opened = true
            shop.Visible = false
            frame.Visible = true
    elseif open then -- If open is true
        open = false
        shop.Visible = true
        frame.Visible = false
    end
end
end)