The reason for starting on the piano is not just because young children cannot reach organ pedals; it's because another layer of coordination is required when playing with the feet as well as the hands. Even on the piano, coordinating the sustaining pedal artistically can take a while to master.
In the UK it is commonly recommended that a pupil reach Grade 5 of the ABRSM piano exams before starting the organ:
http://us.abrsm.org/en/our-exams/piano/piano-grade-5/https://www.youtube.com/watch?v=plA1CQDninsI started the piano at three and a half, though that's unusually early. I discovered the organ when I was eleven, and taught myself for the next seven years (quite likely my parents could not have afforded organ lessons in addition to piano lessons). When I wanted to take the organ as my second instrument at music college, I had lessons from the organist of a local church (a FRCO) for a few months.
What should be covered at what age depends very much on the pupil and on the teacher's priorities. I have always tried to develop reading along with technique, as so many keyboard players are limited by the speed of their reading. The association of a note on the stave with a key on the keyboard needs to be established as early as possible.
Young children can benefit from occasional lessons, without the obligation of a weekly session. You may be able to find a teacher who will offer such a flexible arrangement.
At the risk of stating the obvious, do make sure that the teacher has appropriate qualifications. It ought not to be necessary to mention this, but I live in a town where there are people teaching the piano with no qualification beyond ABRSM Grade 8. The minimum I would look for would be a Licentiate or Associate diploma such as LRAM or ARCM.
This is a huge subject; I hope these few thoughts will be helpful. Please feel welcome to ask further if you need a comment on any particular point.