Why can't I get "v" to change colors?

This code is terrible. Let's fix that, first.

Writing good code makes coding easier. It also means you write fewer bugs.

The first step is always tabbing properly. The next step is good variable names . You have both plr and playa -- neither is spelled correctly¹, nor is plr accurate. Use character and player.

Next, use local variables. Otherwise things will act very unexpectedly².

There's no reason to use hit.Parent["VechicleSeat"]. That exactly the same thing as hit.Parent.VehicleSeat.

pos = Vector3.new(0,1000,0)

function onTouch(hit)
    if hit.Parent.VehicleSeat.SeatWeld then
        for _, part in pairs (hit.Parent:GetChildren()) do
            if part.Name == "paint" then
                local gui = script.Parent.ScreenGui:Clone()
                local character = (hit.Parent.VehicleSeat.SeatWeld.Part1.Parent)
                local player = game.Players:FindFirstChild(character.Name)
                gui.Parent = player.PlayerGui
                hit.Parent.VehicleSeat.Disabled = true
                hit.Anchored = true
                character.Torso.Anchored = true
                character:MoveTo(pos)
                local cam = workspace.CurrentCamera
                cam.FieldOfView = 100
                cam.CameraSubject = hit.Parent
            end

            while wait() do
                local thingy = gui.Color.ToolTip
                print(thingy)
                print(part)
                part.BrickColor = BrickColor.new(Color3.new(thingy))
            end
        end
    end
end

script.Parent.Touched:connect(onTouch)

You have improper error checking. If the object exists, then hit.Parent.VehicleSeat.SeatWeld will pass the if condition, so that if is pointless. If it doesn't exist, it will error. We need to use :FindFirstChild to see if objects exist. Since we would have to do it twice, it might be better to just use early returns instead of ifs -- or better yet, let's use functions!

In addition, your loop tries to change part's BrickColor even when it isn't necessarily a BasePart. You need to check for that.

Moreover, there's no reason to do stuff inside the loop that don't care about part. That means you should cut out the things about gui and cam -- in fact, you're just doing it if there is a child named "paint" does that sound like a use for :FindFirstChild rather than a loop?

function doThing(hit, weld)
    if hit.Parent:FindFirstChild("paint") then
        local gui = script.Parent.ScreenGui:Clone()
        local character = weld.Part1.Parent
        local player = game.Players:FindFirstChild(character.Name)
        gui.Parent = player.PlayerGui
        hit.Parent.VehicleSeat.Disabled = true
        hit.Anchored = true
        character.Torso.Anchored = true
        character:MoveTo(pos)
        local cam = workspace.CurrentCamera
        cam.FieldOfView = 100
        cam.CameraSubject = hit.Parent
        for _, part in pairs (hit.Parent:GetChildren()) do
            if part:IsA("BasePart") then
                while wait() do
                    local thingy = gui.Color.ToolTip
                    print(thingy)
                    print(part)
                    part.BrickColor = BrickColor.new(Color3.new(thingy))
                end
            end
        end
    end
end


function onTouch(hit)
    local vehicleSeat = hit.Parent:FindFirstChild("VehicleSeat")
    if vehicleSeat then
        local weld = vehicleSeat:FindFirstChild("SeatWeld")
        if weld then
            doThing(hit, weld)
        end
    end
end

script.Parent.Touched:connect(onTouch)

Note that you have a while loop that will never stop in the inside of a for loop. That means only the first thing in the for loop will ever get considered.

If you want to just change the color of the part to the color of gui.Color.ToolTip, you should use a Changed event. Either that, or spawn a new thread (but that's messier and slower)

pos = Vector3.new(0,1000,0)

function doThing(hit, weld)
    if hit.Parent:FindFirstChild("paint") then
        local gui = script.Parent.ScreenGui:Clone()
        local character = weld.Part1.Parent
        local player = game.Players:FindFirstChild(character.Name)
        gui.Parent = player.PlayerGui
        hit.Parent.VehicleSeat.Disabled = true
        hit.Anchored = true
        character.Torso.Anchored = true
        character:MoveTo(pos)
        local cam = workspace.CurrentCamera
        cam.FieldOfView = 100
        cam.CameraSubject = hit.Parent
        gui.Color.Changed:connect( property )
            if property == "ToolTip" then
                local color = BrickColor.new( Color3.new( gui.Color.ToolTip ) )
                for _, part in pairs(hit.Parent:GetChildren()) do
                    if part:IsA("BasePart") then
                        part.BrickColor = color
                    end
                end
            end
        end
    end
end

function onTouch(hit)
    local vehicleSeat = hit.Parent:FindFirstChild("VehicleSeat")
    if vehicleSeat then
        local weld = vehicleSeat:FindFirstChild("SeatWeld")
        if weld then
            doThing(hit, weld)
        end
    end
end

script.Parent.Touched:connect(onTouch)

These changes make it easier to read -- which makes it easier to write. It also reduces the bugs, because it's simpler. We've eliminated a loop, reduced levels of indentation by a lot, shortened the main function, etc.