If a person is on a ventilator, and does not have a trachea, how long do hospitals (namely icu's) allow this?

Tracheotomy: I am assuming you are asking about a tracheotomy or tracheostomy. If the person is expected to be on a ventilator long term or has been on the it long term (> 2 weeks) usually a "trach" is done. Sometimes, a trach is done initially if there are airway issues. A trach allows the person to be more comfortable, allows easier nursing care, and prevent future airway complications.

No perfect answer: There is no established time limit. One recent study showed benefit to early tracheostomy in patients who an intensivist identified as needing one eventually. An example would be a high quadripalegic.

Tracheostomy?: If a patient is on (or is anticipated to stay on) a ventilator for a prolonged period, then a tracheostomy may be recommended. In most icus, two weeks is usually the time frame. Sometimes, it's apparent that a tracheostomy is needed after only a few days (patients with severe head injuries, for example); and sometimes it's ok to wait longer than two weeks.