Using NDSolve to find particle trajectory

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}]

Alternative method:

b = {1, 0, 0};
e = {0, 0, 1};
q = 1;
m = 1;

sol = NDSolve[{e + Cross[pos'[t], b] == m/q pos''[t], 
    pos[0] == {0, 0, 0}, pos'[0] == {0, 0, 0}}, pos, {t, 0, 10}, 
   Method -> {"EquationSimplification" -> "Residual"}];
ParametricPlot3D[pos[t] /. sol, {t, 0, 10}, PlotRange -> All]

Another alternative is to package your constant vector parameters as DiscreteVariables. In the OP's case, it necessary only to chnage e since b occurs inside Cross, which will not evaluate until all its arguments are vectors. Note that in the equation we changed e to e[t] and set its value with e[0] == {0, 0, 1}.

b = {1, 0, 0};
(*e={0,0,1};*)
q = 1;
m = 1;

sol = NDSolve[{q*e[t] + 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}, 
    e[0] == {0, 0, 1}}, pos, {t, 0, 10}, DiscreteVariables -> {e}];

ParametricPlot3D[pos[t] /. sol, {t, 0, 10}, PlotRange -> All]