Is it permissible to eat pizza with cheese or vegetables from restaurants that serve pork, knowing that workers may touch the pork and then touch other ingredients, and is it sufficient to ask them to wash their hands and the knife to ensure its lawfulness?

Pork is impure by scholarly consensus. If a worker touches pork with his hands and then places them in moist vegetables, or uses the same knife to cut other foods, those foods become impure. It is not permissible to eat impure food. If you are certain that the worker washed his hands and knife before touching permissible foods, then it is permissible to eat from his food. However, if there is doubt, it is not permissible.