In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, where $w$ divides $|G|$. This is a generalization of an earlier method proposed by Tonchev for constructing self-orthogonal codes from orbit matrices of $2$-designs with a fixed-point-free automorphism of prime order. As an illustration of our method we provide a classification of self-orthogonal codes obtained from the non-fixed parts of the orbit matrices of the symmetric $2$-$(56,11,2)$ designs, some symmetric designs $2$-$(71,15,3)$ (and their residual designs), and some non-symmetric $2$-designs, namely those with parameters $2$-$(15,3,1)$, $2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$, respectively with automorphisms of order $p$, where $p$ is an odd prime. We establish that the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$ are optimal two-weight codes. Further, we construct an optimal binary self-orthogonal $[16,5,8]$ code from the non-fixed part of the orbit matrix of the $2$-$(64,8,1)$ design with respect to an automorphism group of order four.

This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.