A Scaled Fist gains the Draconic Fury class feature at level 3, however a Monk does not gain the Ki Pool class feature until level 4. Does this mean a Scaled Fist cannot use its Draconic Fury ability until level 4?
Correct A Chained Monk that has taken the Scaled Fist archetype cannot use Draconic Fury until they have a Ki Pool, which is at level 4.
It would appear that the Scaled Fist Monk is designed to work best with the Unchained version of the Monk, which gains a Ki Pool at 3rd level, rather than the 4th level of the original Monk. I suspect it was probbaly altered after the fact to work with the original Monk, and the features were left at the levels originally set, despite not working as well.
In theory, it is also possible to have another archetype that moves Ki Pool to an earlier level, and stack it with Scaled Fist. However, the only examples of this I could find (Drinken Master and Ki Mystic) don't stack with Scaled Fist.
It could also be possible to gain a Ki Pool from another source, such as a magic item or other special ability, which would then allow the use of this ability at 3rd level even for the chained monk. I'm not aware of any specific examples of this.
No, for vanilla monks. Yes, for unchained monks
The archetype doesn't seem to change at what level you gain your Ki Pool (normally at 4th for vanilla monks), and the ability makes no mention of changing or replacing your ki pool. Meaning that your ki pool works as normal for monks, and even though you gain Draconic Fury at third level, you will only be able to use it at 4th level.
However, you must pick your element at third level, regardless if you have a ki pool yet or not.
Keep in mind that the developer who wrote this seems to be aware of this difference between the vanilla and unchained monks, as he does mention that for the unchained monk, this replaces a 4th level ability instead of Maneuver Training. The unchained monk doesn't seem to have this problem and can use Draconic Fury at 3rd level normally.
The Pathfinder Society developers also made no changes on this archetype when writing the errata/FAQ for PFS games, which is the only errata we might ever see on player companion books, other than things being reprinted.
Remember that there are other ways to obtain a ki pool before the third level of the Scaled Fist archetype, such as multiclassing as a ninja or teisatsu vigilante, or even the Ki Pool rogue talent. This won't allow you to gain it before the 4th level of your character, but you will be able to use the ability 1 level sooner than you would should you multiclass (5th instead of 6th).