I know that the sum of two independent normally distributed random variables is also a normal random variable, but is this true of other distributions? For example, what probability distribution does the sum of two Von Mises distributions follow?
Yes, this property is true for some but not all other distributions. The distribution for the sum of two independent random variables is the convolution of the individual distributions. Some other distributions for which this holds includes the Poisson, Cauchy, and Gamma distributions (for the Gamma, they must have the same rate parameter). See here a list here.
My answer is incomplete because I don't know if this holds for the von Mises distribution.
Previous answer (retained for completeness)
My answer is incomplete because I don't know what the distribution of the sum of independent Von Mises random variables is, but I do know it is not a Von Mises random variable: the Von Mises distribution has a compact support, and so the sum of two such random variables may therefore fall outside this interval.