How would I disable shift to sprint while a GUI button is activated?

Personally, I would advise you make a BoolValue inside of the player as such

CanSprint = Instance.new("BoolValue", game.Players.LocalPlayer)
CanSprint.Name = "CanSprint"

You should probably do this inside your sprint script, and then make it so that if CanSprint is false, you don't change speed

Sprint Script

local Player = game.Players.LocalPlayer
local Character = Player.Character
local UserInputService = game:GetService("UserInputService")
 
local CanSprint = Instance.new("BoolValue", Player) -- because you have the local Player value
CanSprint.Name = "CanSprint"
CanSprint.Value = true -- Because there is no reason not to have it as true straight away

UserInputService.InputBegan:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift and CanSprint.Value == true then
        Character.Humanoid.WalkSpeed = 35
    end
end)

UserInputService.InputEnded:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift and CanSprint.Value == true then
	--You will notice I added the CanSprint there too, to avoid changing speed after releasing it       
       Character.Humanoid.WalkSpeed = 18
    end
end)

Other Script

local player = game.Players.LocalPlayer
local humanoid = player.Character:WaitForChild('Humanoid')
local char = player.Character
local CanSprint = player:WaiForChild("CanSprint")

script.Parent.MouseButton1Click:connect(function()
        CanSprint.Value = false
        char.Humanoid.WalkSpeed = 6.5
        char.Animate.walk.WalkAnim.AnimationId = "rbxassetid://3670529036"
        char.Animate.run.RunAnim.AnimationId = "rbxassetid://3670529036"
        char.Animate.idle.IdleAnim.AnimationId = "rbxassetid://3670529036"
end)

Of course you would have to toggle it back when you want to be able to run again