Does society play a role in why some people commit suicide?

Certainly. Society sets two standards that apply to suicide: how suicide is accepted by the society at large, and what transgressions are significant enough to warrant suicide as a viable alternative.

Most Western and European cultures take the Judeo-Christian view that suicide is a sin and thus deplorable. As such, social issues that warrant suicide are few and far between, with the greatest leniency in this few given more towards end-of-life scenariOS that involve physical pain.

Japan especially, and many Asian cultures, on the other hand, avow that suicide is, under the correct circumstances, a reasonable solution to ones problems. Ancient Japanese culture in fact listed some forms of ritualized suicide as a means of wiping the slate entirely clean.