Why won't the chat reply on chatted message?

You haven't added the player.Chatted connection line to call the function, also to do this player would need to be the actual player, not the character. Add

player.Chatted:connect(function(msg, recipient, speaker)

instead of

function onChatted(msg, recipient, speaker)

Once you've changed it, it should be like this:

player = game.Players.LocalPlayer -- removed .Character
function chat(text)
    game:GetService("StarterGui"):SetCore("ChatMakeSystemMessage",{
        Text = text;
        Color = Color3.new(255,255,255);
        StrokeColor = Color3.new(0,0,0);
        Font = Enum.Font.SourceSansItalic
    })
end

for i =1,3 do
    if i == 1 then
        chat("Welcome")
    elseif i == 2 then
        chat(player.Name)
        player.Chatted:connect(function(msg, recipient, speaker) -- Made the connection line
            local source = string.lower(speaker.Name) 
            msg = string.lower(msg) 
            if i == 3 then
                if (msg == "no") then -- Message said for the reply
                    chat("rude") -- The reply
                end
            end
        end)
    end
end

I removed the unnecessary for loop and just make it simpler. Here's my finished code:

player = game.Players.LocalPlayer -- removed .Character

function chat(text)
    game:GetService("StarterGui"):SetCore("ChatMakeSystemMessage",{
        Text = text;
        Color = Color3.new(255,255,255);
        StrokeColor = Color3.new(0,0,0);
        Font = Enum.Font.SourceSansItalic
    })
end

chat("Welcome") -- Welcome them
wait(3) -- Wait for it to be removed
chat(player.Name) -- Say their name
wait(3) -- Wait for it to be removed
player.Chatted:connect(function(msg) -- Made the connection line
    msg = string.lower(msg) -- convert to lower case
    if msg == "no" then -- Message said for the reply
        chat("rude") -- The reply
    end
end)