1. Given electromagnetic wave in vacuum

   $E_x = E_0 \sin(kz-\omega t), \qquad B_y = B_0 \sin(kz-\omega t)$

   This is a plane electromagnetic wave propagating in the $+z$ direction.

2. Use the standard relation for EM waves in vacuum

   For an electromagnetic wave in vacuum,

   $E_0 = c B_0$

   where $c$ is the speed of light.

3. Relate $c$ with $\omega$ and $k$

   For a wave,

   $c = \frac{\omega}{k}$

   Therefore,

   $E_0 = \frac{\omega}{k} B_0$

4. Rearrange

   Multiplying both sides by $k$,

   $kE_0 = \omega B_0$

5. Match with the options

   This corresponds to:

   $\boxed{\text{C: } kE_0 = \omega B_0}$

6. Check other options briefly

   • A: $E_0B_0=\omega k$ → dimensionally incorrect.
   • B: $E_0=kB_0$ → incorrect.
   • D: $\omega E_0 = kB_0$ → incorrect.

Hence, the correct option is C.