The tenant was gone for a couple of weeks and over the break, there was a burglary and the burglar had broken into the kitchen window. Who is responsible for repairs to the broken window, the tenant or the landlord? And what are the Missouri laws relevant to this issue?

Unless something is in the lease terms about this, which isn't likely in a standard lease, the owner of the building is responsible, just as he is for a tree hitting the house in a storm, or other damage on the outside caused by factors not involving the tenant. (And inside the house certain systems are the liability of the owner too, such as furnaces and plumbing and wiring).

I'm not sure if there is a specific law covering this but it's common knowledge in all states.
The fact that the tenants were gone is not relevant. They would have to have been gone a long time and left the place abandoned/neglected to be held liable.

This is damage to the structure, therefore the landlord would be responsible and it should be covered by insurance.