There is a well-known construction of the primitive idempotents for the symmetric group over the rationals that is due to Murphy (and possibly Jucys?). It can be found in his paper "A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (1981), 287-297.

To describe this for $1\le k\le n$ let $L_k=(1,k)+\dots+(k,k-1)$ be the $k$th Jucys-Murphy element. Notice that $L_k^*=L_k$ for all $k$. You can check that $L_kL_m=L_mL_k$, for $1\le k,m\le n$. Now for each standard tableau $t$ define
$$ E_t = \prod_{k=1}^n\prod_{c\ne c_k(t)}\frac{L_k-c}{c_k(t)-c}, $$
where $c_k(t)$ is the content of $k$ in $t$ --- that is, $c_k(t)=c-r$ if $k$ appears in row $r$ and column $c$ of $t$. In the second product $c$ runs over all possible possible contents in all standard tableaux of size $n$.

Murphy shows that $\{E_t\mid t \text{ standard}\}$ is a complete set of pairwise orthogonal idempotents in $\mathbb{Q}S_n$. In fact, this is almost immediate if you just consider how these elements act on the seminormal representations of the symmetric group. For a more modern account see Okounkov and Vershik's paper "A new approach to representation theory of symmetric groups", Selecta Math. (N.S.), 2 (1996), 581–605.

By construction, $E_t^*=E_t$ so these idempotents have the property that you want.

For calculations these idempotents can be constructed recursively. They become a mess, however, if you expand them in the permutation basis of the group algebra, so you should try and avoid this if possible.