The following is NOT recommended for a low-level start game. It is presented as a level 20 build. There are ways to modify it to be more playable at low levels, but it would be weaker at high levels.
Gunslinger 5 (Pistolero): Grants Firearms, Grit, some Deeds and Dex-to-Damage.
Ranger 2 (Trophy Hunter): Extra Grit and a 1st level Gunslinger Deed.
Summoner 12 (Synthesist): Replace our Physical scores, grow a LOT of arms.
Something! 1: Free space! This is actually a level 19 build! Options will be explored later! For build purposes we shall assume a level of Fighter! (for the bonus feat!)
Race: Halfling, small, -2 Str +2 Dex, +2 Cha (Halfling chosen to allow small-Eidolon-Form)
Gunslinger 5 (Pistolero)
Ranger 2 (Trophy Hunter)
Summoner 12 (Synthesist)
Something! 1 (who knows!)
Sample 15-point buy:
Str 9 (11-2race)
Dex 16 (13+2race+1level)
Wis 14 (13+1level)
Cha 18 (13+2race+3level)
8d10+12d8 HD (avg 98 +40 from Con) + 9d10+Con from Eidolon.
All saves will be medium/great
Gunslinger 5 gets us Guns, Grit and Gore (Dex-to-Damage that is)
Ranger 2 is to improve our Grit pool and grab the Mysterious Stranger's Focused Aim deed.
Summoner 12 (Synthesist) The Eidolon's physical stats will replace our own (hence our poor Str/Dex scores), we will be using a Small Serpentine Eidolon for the net +2 to-hit +0 damage (-1 Weapon size, +1 Dex-to-damage). Eidolon-form stats after all bonuses/penalties are Str 13, Dex 24, Con 12. We have 16 Evolution points, most will be spent on adding Arms, the remainder will be spent on utility powers or boosting Dex.
Something! 1. The next levels in each of our 3 chosen classes provide minimal gains. Possibilities:
Feats: 11 (10 levels, 1 Gunslinger) these are listed in no particular order
, and are only numbered/labeled to keep track of them easier.
1 Rapid Reload
7 Point-Blank Shot
11 Two/Multi-Weapon Fighting (Multi specifically replaces Two-weap for creatures with 3+ arms, the problem is that we only have 2 arms in our natural state, so we need to have 15 Dex to take Two-Weap, when Multi only requires 13 Dex)
13 Deadly Aim
15 Precise Shot
17 Deft Shootist Deed
19 Leaping Shot Deed
(G) Improved Critical (Keen is not an option on firearms)
(F) Hammer the Gap
We shall assume your Eidolon-form has 12 arms total, each holding a pistol. Most of your weapons will probably have minimal upgrades based on what you can afford, recommend Distance on most/all of them, and buy a wide variety of abilities. 1 of your weapons can be a 'dud' gun with no abilities, after the first shot you drop it to free up that hand for reloading.
Damage Calculation (assume all pistols are simple +1's, and that the Eidolon has 26 Dex):
12 attacks, each with...
To-hit: +21 (17bab, 8dex, 1weapon, 1size, -5deadly aim, +1point-blank, -2multiweapon (assumes Pistols are Light weapons))
Damage: 1d6+24 (1d6+1weapon, +10deadly aim, +8Dex, +4Cha, +1point-blank)
Most will hit, and 1 of those should be a Critical for x4 damage, and will regain the Grit point spent on Focused Aim. You can also spend a Grit to use Leaping Shot Deed and move before/after/during your attacks.
If all of them hit, with 1 Critical, you will be doing around 400 damage each round! (somebody double check please)
Hammer the Gap damage is hard to calculate in. You only gain the bonus on 'consecutive' hits, so any Miss will reset the counter, also, the bonus damage IS multiplied on a Critical.
If all 12 attacks hit with NO Critical, you'd gain +66 damage from Hammer the Gap!
Round 2 of combat might be trickier, as you will need to reload a dozen guns (but then again, most things will be dead so...)
HIGHLY recommend you make use of Alchemical Cartridges to get free reloads (you still need 1 free hand).
Final notes: The presented build is merely 1 option, feel free to adjust class levels up/down.
Pumping both Wis and Cha is inefficient when your Grit pool 'can' run entirely on Cha with Mysterious Gunslinger. In that case you should use the Ranger levels to grab the Quick Clear deed, and then you only need the 1 level of Gunslinger. With the 4 extra levels, Fighter would be a good choice for the Feats, and you can pick up Weapon Specialization to help make up for the loss of Dex-to-Damage.