Learn English – Not understood meaning of sentence

The confusion is probably because this is an archaic construction. It is called genitive case. Teachers is not the subject of the sentence, although it appears to be because it comes first. The gerund phrase coming to our schools is the real subject, and it belongs to the teachers because it is an action they will undertake.

To be grammatically correct, the sentence should have a possessive apostrophe after teachers: Your teachers' coming to our schools would fulfill the contract.

Other ways to phrase the same idea:

• Your teachers, by coming to our schools, would fulfill the contract.
• By coming to our schools, your teachers would fulfill the contract.
• To fulfill the contract, your teachers must come to our schools.
• Coming to our schools is how your teachers would fulfill the contract.
• The contract will be fulfilled when your teachers come to our schools.
• etc. etc.