Is it permissible to touch women's bodies to draw blood from them, given that the questioner works in a medical center, there is no male alternative to perform the task, and he is in urgent need of the job?

It is not permissible for a man to touch a woman who is a stranger to him, due to the saying of the Prophet (peace be upon him): "For one of you to be stabbed in the head with an iron needle is better for him than to touch a woman who is not lawful for him." Touching becomes even more forbidden if it involves feeling the hand or elbow. Being in a trial period does not permit you to violate this boundary. You should try to communicate with your supervisor and explain to him that your religion prevents you from touching women, especially since there are nurses who perform this task. If there is no alternative and you are in genuine need of this job, you can continue with the training and avoid touching women as much as possible. If you are forced to do so, limit it to what is absolutely necessary, while lowering your gaze.