The simplest approach would be to make the puppeteer a summoner unchained with an inevitable eidolon. While the eidolon is technically an outsider, the inevitable subtype makes it “construct-like.” You could also pretty easily use the summoner’s summon monster ability refluffed as puppets. This is an easy solution, but the summons are not actually constructs (and making them react like constructs would dramatically improve their potency, since constructs have many powerful immunities).
For the actual construction of magic dolls or puppets, the Craft Construct feat is the relevant way to do so, but Caster Level 5th requirement is a bit tough. If you want a lower-level puppeteer, I recommend porting a concept from the 3.5 Eberron Campaign Setting: there, the artificer class got “Craft Homunculus” as a class feature; this was equal to Craft Construct except that
The artificer got it early, and
It only allowed the construction of homunculi, which are much smaller and weaker than other constructs. Eberron Campaign Setting and other Eberron supplements also supplied quite a selection of homunculi that one could build.
So my suggestion here is to give this character a Craft Homunculus ability, which is a limited form of Craft Construct but available at a lower level. Furthermore, this ability would waive the spellcasting requirements for homunculi creation: primarily, this removes the need for arcane eye, a 4th-level spell. For example, you could add this feat to the game:
You can create homunculi, small construct creatures.
Caster Level 3rd
You can create homunculi, and you can supply magic which replaces the required spells of creating a homunculus by making a Use Magic Device check of 15 + twice the replaced spell’s level (a failed check on any of the required spells means that the day’s work is wasted). All other requirements of homunculus-creation must be met as normal. The act of animating a homunculus takes one day for each 1,000 gp in its market price. To create a homunculus, you must use up raw materials costing half of its base price, plus the full cost of the basic body created for the construct. A newly created homunculus has average hit points for its Hit Dice.
This is very similar to the artificer’s ability to create homunculi; the 3.5 artificer was an overpowered class, but not because of the homunculi. I have played in games where an artificer decided to focus on the homuculi, and that focus tended to result in a better game than games where an artificer focused on some of the more powerful things they could do.
Alternatively, and now we’re getting into territory that I have not yet personally tried, this could be part of an archetype for the summoner, one that also perhaps gives the base evolutions of the inevitable eidolon to any outsider summoned with the summoner’s summon monster ability. In return, the summon monster abilities of the summoner could be slowed one step: the summoner gains no ability to summon a monster at 1st level, and only gains summon monster I at 3rd level, and so on.
Either way, if you have Eberron Campaign Setting or other Eberron books, porting the homunculi to Pathfinder would expand the puppeteer’s repertoire in a reasonably-safe way. Alternatively, classifying some of the lower-CR constructs that mxyzplk mentions (guardian doll, soulbound doll) would also likely be fair.
Another approach I have used for a puppeteer, which you will probably want to use in conjunction with one of the options above, is to invent a new “weapon” type, the doll.
The doll is a ranged weapon with a 15-ft. range increment and a 30-ft. max range. The weapon consists of a Tiny (for a Medium wielder; in general two size categories smaller than the wielder) automaton which moves to the target and attacks, and then returns. Thus, there must be a path for the doll to walk to the target, but the doll can go around obstacles so long as the path is not longer than the weapon’s range. Cover and concealment are determined from the wielder’s position, however, making the weapon difficult to use around (opaque) corners. The weapon’s damage is equal to the weapon used by the automaton. Most dolls are capable only of using a single light or one-handed melee weapon appropriate to their size.
The doll does not provoke attacks of opportunity for its movement when used to attack. It is not an independent creature and is targeted as an object.
Therefore, a doll sized for a Medium wielder would wield a Tiny weapon, e.g. a tiny longsword that deals 1d4 slashing damage with 19-20/×2 criticals.