Realizing a subgroup of a Lie group as a stabilizer subgroup

You need $H$ to be closed. The Mostow-Palais theorem (1,2,3) then gives what you want -- with "equal" in place of "locally isomorphic", but with a possibly reducible representation. I'm not aware of conditions ensuring that the representation can be chosen irreducible.

(1): http://en.wikipedia.org/wiki/Mostow-Palais_theorem

(2): http://books.google.com/books?id=oCO0xOzNLhAC&pg=PA373

(3): http://books.google.com/books?id=yqbocEpFdyQC&pg=PA104

Though it's not so readily available online, there is an elementary textbook treatment of the basic theory here in Representations of Compact Lie Groups by Brocker and tom Dieck (GTM 98, Springer, 1985).

1) As they note early in the book, the isotropy group $H$ of a point $v$ must be closed in an arbitrary topological group $G$ acting continuously: here $H$ is the inverse image of $v$ under the (continuous) orbit map $g \mapsto g \cdot v$.

2) As a corollary of the Peter-Weyl theorem (using representative functions), they derive easily in III, (4.6) for any compact Lie group $G$ (not necessarily semisimple): Every closed subgroup $H$ of $G$ appears as the isotropy group of an element of some $G$-module.

3) Unfortunately, since all of this theory is somewhat abstract, it doesn't seem to shed light on your question about finding an irreducible representation. However, you do have complete reducibility here of all (necessariy finite-dimensional) representations, which are usually studied in the essentially equivalent (complexified) Lie algebra setting. So if there is a counterexample it would probably be best located there. It seems hard to compute directly with group elements and group representations, but for example Willem de Graaf has worked extensively with computer algorithms for both real and complex Lie algebras.

P.S. Though Dan Mostow (who just turned 90) was on my thesis committee, I don't think you need to get into his more general results involving group actions on manifolds.