Those ?NumericQ in the definition of f interfere with symbolic computation. Also, if you calculate the derivative with D[f[x, a], x], the variable x will become numerical too soon, before the symbolic derivative is calculated. One way to solve the problem is to use Derivative, which does not use a symbolic variable name:

f[x_, a_] := Sin[a x]
d[x_?NumericQ] := NIntegrate[Derivative[1, 0][f][x, a], {a, 0, 1}]

Another way, which localizes the symbolic variable:

f[x_, a_] := Sin[a x]
d[x_?NumericQ] :=
 Module[{y},
  NIntegrate[D[f[y, a], y] /. y -> x, {a, 0, 1}]]