What is the ruling on using water from the restaurant, which is designated for work, to wash a personal car? And is it obligatory to inform the person in charge, or to give charity on their behalf? And how can one give charity on their behalf when feeling embarrassed to inform them?

If the water is something customarily tolerated for use without charge, then there is no blame upon you, and nothing is required of you. However, if the water is something not usually tolerated without payment, then you must repent to Allah and return the value of what you took to the restaurant owners, unless they forgive you. Otherwise, the value should be returned to them without informing them. Charity is not sufficient as long as you are able to return the right.