The main problem is that your pos is not seen as a 3D vector.
The cross product is therefore interpreted as a scalar:
q*Cross[D[pos[t], t], b]
when adding this to the vector q.e this 'scalar' term is added to each of the vector components:
q*e + q*Cross[D[pos[t], t], b]
This won't work, instead do:
b = {1, 0, 0};
e = {0, 0, 1};
q = 1;
m = 1;
Define pos as a 3D vector. Also take more time than a single second:
ClearAll[pos]
pos[t_] = {px[t], py[t], pz[t]};
sol = NDSolve[
{
q*e + q*Cross[D[pos[t], t], b] == m D[pos[t], {t, 2}],
pos[0] == {0, 0, 0},
(D[pos[t], t] /. t -> 0) == {0, 0, 0}
}, pos[t], {t, 0, 20}]
ParametricPlot3D[Evaluate[pos[t] /. sol], {t, 0, 20}]
