Do you need to write a resignation letter if on probation period at work?

Put it in writing. Tell them the date that you are leaving and ensure you ask for all outstanding wages to be paid. You don't have to give a reason, but you can say "due to not being issued with a contract" or "the probation period has shown me that my career lies elsewhere" .Keep a copy.

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Also, make sure you thank them for the opportunity. It costs nothing to be nice and you never know when you will come across these people again so make sure you leave on a good note (thank you for the opportunity, have enjoyed my time here, some high level reason for not wanting to continue, wish them luck with their future endeavours)