NIntegrate of a vector-valued InterpolatingFunction gives "not numerical"

You can use Indexed to access the components separately:

NIntegrate[Indexed[f[t], 1], {t, 0, 3}]
NIntegrate[Indexed[f[t], 2], {t, 0, 3}]
(*
  0.75
  1.5
*)

Integrate will antidifferentiate an InterpolatingFunction. You can then subtract its values at the end points.

af = Head@Integrate[f[t], t];
af[3] - af[0]
(*  {3/4, 3/2}  *)

You can also write your own integration rule to plug into NIntegrate, but that takes a little work.

You can use NDSolve to integrate your interpolating function:

f = Interpolation[{{0,{1,1}},{1,{0,0}},{2,{0,1}},{3,{1,0}}}];

NDSolveValue[{y'[x] == f[x], y[0] == {0, 0}}, y[3], {x, 0, 3}]

{0.75, 1.5}

Here is a more complicated integrand:

NDSolveValue[{y'[x] == f[x]^2 Sin[x], y[0] == {0, 0}}, y[3], {x, 0, 3}]

{0.148276, 0.887794}

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