What should you do if your wife is ill and cannot accompany you when applying for a marriage visa in Thailand?

Get a statement from her doctors that she is not able to travel to the Immigration Office to support you. Immigration can make her picture during the home visit.

Postpone it till she gets better.

That is a tricky question. The general 'rule' is that you need the reason you're applying for a visa/extension to accompany you to the immigration office (for marriage you need your wife and for raising kids you need at least one of the children)

You can certainly meet the documentation requirements without your wife doing much more than signing the appropriate copies with her name on them. BUT I think you're not gonna get the immigration office to accept your 90 day Non-O visa application without her being there OR without having documentation from a hospital/treating physican that she physically can't go with you.

I would certainly go talk to your immigration office and see what options you have (if any).. And do it well before you need to apply for the 90 day Non-O visa