1. **State the problem:** Find the angle between $0$ and $2\pi$ that is coterminal with $-\frac{5\pi}{6}$.\n\n2. **Recall the concept:** Two angles are coterminal if they differ by an integer multiple of $2\pi$. This means we can add or subtract $2\pi$ to the given angle to find an equivalent angle within the desired interval.\n\n3. **Apply the formula:** To find a coterminal angle $\theta$ in $[0, 2\pi)$, use\n$$\theta = -\frac{5\pi}{6} + 2\pi k$$\nwhere $k$ is an integer chosen so that $\theta$ lies between $0$ and $2\pi$.\n\n4. **Calculate:** Start with $k=1$:\n$$\theta = -\frac{5\pi}{6} + 2\pi = -\frac{5\pi}{6} + \frac{12\pi}{6} = \frac{7\pi}{6}$$\nSince $\frac{7\pi}{6}$ is between $0$ and $2\pi$, this is the coterminal angle we want.\n\n5. **Check other options:**\n- $\frac{13\pi}{5} \approx 2.6\pi$ (greater than $2\pi$)\n- $-\frac{7\pi}{6}$ is negative, not in $[0, 2\pi)$\n- $\frac{3\pi}{2}$ is coterminal with $\frac{3\pi}{2}$, but not with $-\frac{5\pi}{6}$\n\n**Final answer:** $\boxed{\frac{7\pi}{6}}$ (Option A)