Many parents are surprised when I tell then that you need your adult teeth to play the clarinet. Yes, you need your teeth - especially the front top and bottom teeth. You also need to be big enough to hold the instrument and reach the holes and keys. So with those things in mind, the youngest age to start learning the clarinet successfully is age 9.
In general, there are not smaller clarinets to start learning on, as there are with violins and cellos. There is a "C" clarinet which is a little smaller and easier to handle, but is it not the same pitch as the clarinets that are played in school bands, etc. There is also an even smaller "E Flat" clarinet, but this instrument is more difficult to play and not something that a young child would be successful starting on. There are a couple of "kinder clarinets", but it's a different learning experience that I don't teach.
There are always exceptions though. I once had a student begin lessons at age 7. She did very well and continued into adulthood as a fine player.
Students of any age who begin with learning proper fundamentals of technique and musicality have a great learning experience, enjoyment and success!