It was proved by W. Feit and G. Seitz that there are (at most) five non-Abelian simple groups which are not alternating and may occurs as composition factors of groups with integral character table (to use your terminology). There is a reference to their paper in the 2008 paper of J.G. Thompson "On composition factors of rational finite groups" ( Journal of Algebra, 319 (2008), 558-594.

In his paper, Thompson proves that if a cyclic group of order $p$ occurs as a composition factor of a rational finite group, then $p \leq 11$, and expresses the belief that the correct bound may be $p \leq 5$.

Work on solvable rational groups has been done by R. Gow, and later by P. Hegedus.

It was proved by W. Feit and G. Seitz that there are (at most) five non-Abelian simple groups which are not alternating and may occurs as composition factors of groups with integral character table (to use your terminology). There is a reference to their paper in the 2008 paper of J.G. Thompson "On composition factors of rational finite groups" ( Journal of Algebra, 319 (2008), 558-594.

In his paper, Thompson proves that if a cyclic group of order $p$ occurs as a composition factor of a rational finite group, then $p \leq 11$, and expresses the belief that the correct bound may be $p \leq 5$.

Work on solvable rational groups has been done by R. Gow, and later by P. Hegedus.

Later edit: A more general supply of rational finite groups than symmetric groups is provided by finite real reflection groups, which usually have all complex characters rational-valued.