As suggested in [107, p. 752], there is a connection between polynomials of binomial type and compound Poisson processes. Two different approaches can be found in [12,114].
A connection of polynomials of binomial type with renewal sequences can be found in [113].
Probabilistic aspects of Lagrange inversion and polynomials of binomial type can be found in [115].
Various probabilistic representations of Sheffer polynomials can be found in [22].
The following papers do not actually use umbral calculus, but provide interesting information on probabilistic aspect of generalized Appell polynomials [30,31,32,61].