How detect key presses through serverscript?

You can only detect user input through local scripts as they are running on the player's computer; unlike the server, which has no relation with any of the clients.

However, there's an easy way around this, and that's by using a Remote Event.

A Remote Event allows communication between the server and the client, providing a one-way message allowing Server Scripts to call code in LocalScripts and vice-versa. Here is the documentation if you'd like to read more about Remote Events.

It's also recommended to use UserInputService rather than the KeyDown event, as the KeyDown event is deprecated.

Local Script

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()
local char = script.Parent
local inputservice= game:GetService("UserInputService")

local RemoteEvent = game.ReplicatedStorage.RemoteEvent

inputservice.InputBegan:Connect(function(Input, GameProcessed)
	if GameProcessed then return end

	local key = Input.KeyCode
	if key == Enum.KeyCode.V then
		if char.Role.Value == "Vampire" and char.BeastMode.Value == false then
			RemoteEvent:FireServer() -- Pass parameters if you need
			print("Beast Mode!")
		end
	end
end)

Server Script

local RemoteEvent = game.ReplicatedStorage.RemoteEvent

RemoteEvent.OnServerEvent:Connect(function(player)
	
	local char = player.Character

	if char.Role.Value == "Vampire" and char.BeastMode.Value == false then
		char.BeastMode.Value = true
		char.Head.Color = Color3.new(0.74902, 0.74902, 0.74902)
		char.Torso.Color = Color3.new(0.74902, 0.74902, 0.74902)
		char["Left Arm"].Color = Color3.new(0.74902, 0.74902, 0.74902)
		char["Left Leg"].Color = Color3.new(0.74902, 0.74902, 0.74902)
		char["Right Arm"].Color = Color3.new(0.74902, 0.74902, 0.74902)
		char["Right Leg"].Color = Color3.new(0.74902, 0.74902, 0.74902)
	end
end)