How's this?
OmegaIntegrate = Evaluate@Integrate[Omega[a], a]
Animate[Graphics3D[{Cuboid[{-20, -20, -1.01}, {20, 20, -1}],
GeometricTransformation[{Green,
Tube[{{0, 0, 0}, {0, 0, 0.15}}, {1.25, 0}],
Cylinder[{{0, 0, 0}, {0, 0, 1}}, 1/4], Blue,
Cylinder[{{0, 0, 0}, {0, 1, 0}}, 1/4]},
RotationTransform[Block[{a = t}, OmegaIntegrate], {0, 0, 1}]]},
Boxed -> False, SphericalRegion -> True, PlotRange -> 3,
PlotRange -> 3], {t, 1, 10, 0.04}, AnimationRunning -> False,
SaveDefinitions -> True]
Since you defined OmegaIntegrate as a variable, you can't call it as a function. You have to use Block to define the value of t to use.
I changed t to a in OmegaIntegrate because Block[{t=t},...] wouldn't work.