How to show a time-dependent vector (velocity) with Manipulate?

You can do it thus:

Manipulate[
 Show[trayectoria, 
  Graphics3D[Arrow[{pos, pos + vel} /. t -> α]]], {α, 0,
   10 Pi}]

By defining your kinematics equations as top-level expressions (rather than as functions of the relevant quantities), you invite scoping problems. I recommend working with functions and making the Manipulate expression self-contained. Implementing that will simplify your work as well saving you worry about variable scoping issues. Here is how I would do it.

With[{ρ = 1, ω = 1/2, a = 1/5},
  Manipulate[
    Show[
      Graphics3D[
        {Thick, Arrow[{posicion[t], posicion[t] + velocidad[t]}]}], 
      ParametricPlot3D[posicion[u], {u, 0, 10 Pi}, PlotStyle -> Red],
      Boxed -> False],
    {t, 0, 10 Pi},
    Initialization :> (
      posicion[t_] := {ρ Cos[ω t], ρ Sin[ω t], a t}; 
      velocidad[t_] = D[posicion[t], t])]]

Short and simple, no?