In arbitrary characteristic they are locally complete intersection, hence Cohen-Macaulay: this is in SGA3, Exposé VII$_B$, Cor. 5.5.1.

In arbitrary characteristic they are locally complete intersection, hence Cohen-Macaulay: this is in SGA3, Exposé VII$_B$, Cor. 5.5.1 or Demazure-Gabriel, Groupes algébriques, chap. III, §3, n°6. This is due to the structure theorem which says that (after extension to an algebraic closure of the base field $k$) the completed local ring of the unit element is of the form $k[[t_1,\dots,t_n]](X_1,\dots,X_r)/(X_1^{p^{n_1},\dots,X_r^{p^{n_r})$, a local complete intersection.