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.
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.
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.