Representations of finite commutative band semigroups

Firstly, if you're only interested in finite semigroups, I suggest amending the title of the question ;)

Commutative semigroups in which ever element is idempotent are called semilattices, and are a special case of the more general notion of inverse semigroup. They have been much studied, although their infinite-dimensional representation theory seems slightly less well mapped out. In any case, the correspondence you describe between semilattices as semigroups and semilattices as certain kinds of poset is indeed well known.

If you're only interested in finite semilattices, then I think the paper to look at is one by Solomon where he gives explicit formulas for the central idempotents that generate the semigroup ring of a given finite (semi)lattice. Unfortunately I've forgotten the exact reference, but I think similar versions or improvements are discussed in

C. Greene, On the M"obius algebra of a partially ordered set, Adv. in Math. 10 (1973), 177--187.

G. Etienne, On the M"obius algebra of geometric lattices, Eur. J. Combin. 19 (1998), 921--933.

(Off the top of my head, for representations of more general finite semigroups, see recent work of Benjamin Steinberg, which is where I first became aware of some of the older work.)

I would have thought that the representation theory of rectangular bands should actually be much easier than that of general bands, since every rectangular band can be rewritten as $L times R$ for index sets $L, R$ and with multiplication defined by $(l_1,r_1)cdot (l_2,r_2)=(l_1,r_2)$.