The NSW Home Building Act requires any residential building works over $20K to have a contract that meets the requirements of the legislation and that a certificate of insurance be provided (home warranty insurance). Similar provisions apply in each state. Note also the contractor cannot require more than a 10% deposit.
Even smaller jobs require a basic written contract.
Under the Act residential building works includes repairs (and also things such as plumbing, electrical etc.)
Presuming the company has a licence (have you checked, including whether there are any matters recorded against them), which they are also required to have to undertake this kind of work, they would know this, and that to not have a contract in place and provide a certificate of insurance is a breach of the law.
If I were on the EC I would be very alarmed if a tradesperson refused to enter into a contract, because the OC would have no legal protection, and if something went wrong it would have no redress. In the case of the EC members, and taking into account that you have checked with Fair Trading and been told, you might well be liable if something went wrong.
Your strata manager should know this and advised you not to proceed without a contract.
i know what it is like to have difficulty finding tradesmen to do a job, but I have also been on the other side where things went badly wrong on a project. Because we had made sure that all the requirements were met, we didn’t suffer any financial loss and were able to claim on the HWI.