Estimating kappa of von Mises distribution

Yes, the Von-Mises distribution family is an exponential family, so you can find the maximum likelihood estimate of its parameters as you would for any exponential family: set the expectation parameters to the average of the sufficient statistics $T(x) = x$ whose magnitude we'll call $bar r$. You'll have to convert these parameters to your parametrization after to get $kappa$. See @mic's answer for the equation.

Just in case you're wondering how you implement @mic's solution in Python: I would use scipy.optimize to find the root of your function: the ratio of Bessel functions minus $bar r$.

According to Banerjee et al., Clustering on the Unit Hypersphere using von Mises-Fisher Distributions (J. Mach. Learning Res. 6 (2005)), you can estimate the von Mises-Fisher parameters $mu$ and $kappa$ with maximum likelihood.

Let $x_i$ be the $n$ points in dimension $d$ from your sample.

Let $r = sum_i x_i$.

Let $overline{r} = frac{||r||_2}{n}$ (the Euclidean distance from the barycenter to the origin). Then

$$hat{mu} = frac{r}{||r||_2}$$

(the unit vector in the direction of the barycenter) and

$$hat{kappa} approx frac{overline{r}d - overline{r}^3}{1 - overline{r}^2}$$

approximates the Maximum Likelihood estimate, which to be found exactly is obtained (numerically) as the solution to

$$I_{d/2}(kappa) / I_{d/2-1}(kappa) = overline{r}.$$

$I_m$ is the modified Bessel function of the first kind of order $m$. The approximation can be used as the starting point for Newton-Raphson iteration--but the authors point out that for "high-dimensional" data this can be "quite slow" compared to the cost of computing just the approximate value.

Check out the est.kappa() function in the CircStats package for R:

Computes the maximum likelihood estimate of kappa, the concentration parameter of a von Mises distribution, given a set of angular measurements.