Can the median be found for an open ended distribution?

Yes, if it is a mathematically well behaved distribution.

For a symmetric distribution, such as the Gaussian (Normal) it is the same as the mean.

For the Standard Normal or the Student's t the median is 0.

For the exponential distribution, with parameter lambda, the median is ln(2)/lambda. (ln = natural logarithm.]

A chi-squared distribution with k degrees of freedom has a median approximated by k*(1 - 2/9k)^3.

For a Beta(a, b) distribution the median depends on the incomplete Beta function but if a and b >1, then it can be approximated by (a - 1/3)/(a + b - 2/3).

For a Weibull distribution, with scale parameter L and shape parameter K, it is L*(ln2)^(1/K).