Does Missour Law state that an employer must pay his employees on the established pay date?

It doesn't appear that such a requirement exists per se. Here are the only two laws I could find regarding required paydays in Missouri.
http://www.moga.state.mo.us/statutes/C200-299/2900000080.HTM
http://www.moga.state.mo.us/statutes/C200-299/2900000090.HTM

Why are you asking (if I can't guess)?

My husband's employer did not pay on payday because his bookeeper was sick? I didn't think it was lawful for an employer not to pay his employees on the date the company has established as their pay day.

If state laws don't require an employer to establish and stick to pay dates, then the employer merely must pay within whatever time frame the state has established, if any. Some states have no laws dictating how long after the end of a pay period employees must be paid, but Missouri does. So long as the pay was received within that time frame, no law was broken.

My husband's employer did not pay on payday because his bookeeper was sick? I didn't think it was lawful for an employer not to pay his employees on the date the company has established as their pay day. In some states, it isn't; in your state, there is no law one way or the other except, as ScottB said, that employees must be paid as I described in my original response.