Praise be to Allah.
Writing down the contracts and agreements that people make among themselves is a way of confirming and formalizing them, but it is not a condition for the validity of the contract. The marriage contract may be made verbally by means of an offer on the part of the bride’s wali or guardian (e.g. “I marry my daughter to you”) and acceptance on the part of the groom (e.g. “I accept”). Writing it down is not a condition, but it does serve to document and confirm the contract, which is a good thing, especially nowadays. And Allaah is the source of help.