Yes, since it is a transform bringing the threshold from the cumulative distribution space into the (cumulative) probability space. p[k] = sigma[k - 1] - sigma[k] calculates the differences between those probabilities, which we put a uniform Dirichlet prior (\alpha = 1) on.
I think
sigma = inv_logit(phi - c)
is the only line that needs changing for a probit model. Into:
sigma = Phi(phi - c)
EDIT: Fix in code, into Phi instead of inv_Phi. inv_logit is the CDF of the logistic distribution, Phi is the CDF of the (standard) normal distribution. A little confusing.