Erdmann's article <a href="https://academic.oup.com/qjmath/article/44/1/17/1549260">Schur algebras of finite type</a> shows that $S(n,r)$ has finite representation type in prime charateristic $p$ if and only if $n=2$ and $r < p^2$ or $n \ge 3$ and $r \le 2p$ or $p=2$, $n=2$ and $r=5$ or $7$. In these cases the quiver and relations for (the basic algebra Morita equivalent to) each block are found explicitly. Quivers and relations are also found for some blocks of Schur algebras of infinite representation type. For example, Proposition 5.2 gives a basic algebra Morita equivalent to the principal block of $S(2,p^2)$ or $S(2,p^2+1)$ when $p > 2$. There are further results in Section 5 on blocks of $S(3,r)$ of infinite type.

More recently <a href="https://link.springer.com/content/pdf/10.1007/PL00004755.pdf">Doty, Erdmann, Martin and Nakano</a> have classified all the tame Schur algebras. As one would expect, their paper gives some information about the Ext quivers. For example, see  page 153 for the quiver for the basic algebra of $S(2,6)$ in characteristic $2$: it has wild representation type.