Attempt to index nil with 'leaderstats'?

Also if you only sell with a local script it won't permanently change since its only local there a big difference between local and server, It works with filtering enabled, Basically if you don't know what a remote event does, You use a remote event for filtering enabled, For example, you want to add leader stats to a player, you can fire it from a local script to a server plus sending data like the player to the server, So it will be permanent because everything changed for a server will be for everyone, Hope this helped!

How to do it! :

add in a remote event

Put this script as a local script into your Gui

script.Parent.MouseButton1Click:Connect(function()
    game.ReplicatedStorage.RemoteEvent:FireServer()
end)

add a server script into the server script service

game.ReplicatedStorage.RemoteEvent.OnServerEvent:Connect(function(player) -- Always when using remote events or remote functions add player into it

    player.leaderstats.Coins.Value = player.leaderstats.Gems.Value * 5
    player.leaderstats.Gems.Value = 0

    print(player.Name.." has sold their gems")
end

This could be done in a lot of different ways, but here is my take on it. It might not be the most secure way of doing this, but I am not an expert and I just want to help out another fellow member of this community.

Inside the scripts I provide some explanations to how this system works.

In ReplicatedStorage I want you to put a RemoteEvent and name it "sellGems". Because of Roblox's way of keeping your games more secure, we need this for the client's script to be able to contact the server script which will then change the leaderstats values.

In the ServerScriptService I want you to put a normal script, name it whatever you want but it doesn't really matter.

In your gui inside of your button I want you to put a localscript, name doesn't matter here either.

Now put this code in the local script.

--Variables--
local rs = game:GetService("ReplicatedStorage") 
local sellGemsEvent = rs:WaitForChild("sellGems") 


--Functions--
function onClick() 
    sellGemsEvent:FireServer() --This is gonna send an event signal via our new event to the server-side script in ServerScriptService.
end


--Events--
script.Parent.MouseButton1Click:Connect(onClick) --Runs the function onClick() we made earlier.

Put this code in the server-side script.

--Variables--
local rs = game:GetService("ReplicatedStorage")
local event = rs:WaitForChild("sellGems")

--Functions--
function sellGems(plr) --plr is the one who fired the event, which triggered this function.
    plr.leaderstats.Coins.Value = plr.leaderstats.Coins.Value * 5 --Like what you wrote in your script, but now it's server-side so it should work.
    plr.leaderstats.Gems.Value = 0 --Like what you wrote in your script, but now it's server-side so it should work.
end


--Events--
event.OnServerEvent:Connect(sellGems) --Runs the function called sellGems(). It also provides information on who sent the event.

Like I said, read the comments on the scripts where I explain what the code does. Hopefully this works! If you still get errors, check if the leaderstats work correctly and if your values are correctly named and capitalized.

script.Parent.MouseButton1Click:Connect(function()
    local player = game.Players.LocalPlayer
    player.leaderstats.Coins.Value = player.leaderstats.Gems.Value * 5

    player.leaderstats.Gems.Value = 0

    print(player.Name.." has sold there gems")
end)

make it local script otherwise it will not work in any conditions done