(Reminder: games sold through on demand ends September 2017. Get them while you still can)

I made a compiled list of all the Xbox 360 shmups that are playable on a North American Xbox 360 without any hacking or mods to the console. I did not list rail shooters or arena shooters. Let me know if I am missing any:

Disc Based Region Free Shmups:
Bullet Soul (J)
Bullet Soul Infinite Burst (J)
Dodonpachi Resurrection Deluxe Ed (PAL) <does not work on NTS-J console>
Dodonpachi Saidaioujou (J)
Eschatos (J)
Espgaluda II Black Label (J) <standard edition is region free>
Game Center Love Plus Pengo (J)
Ginga Force (J)
Mushi Mushi Pork & Pink Sweets (J)
Mushihami-Sama Futari Ver 1.5 (J)
Shooting Love 10th Anniversary (J)

Disc Based North American Releases:
Akai Katana (NTSC-U)
Deathsmiles (NTSC-U)
Otomedius Excellent (NTSC-U)
Raiden Fighter Aces (NTSC-U)
Raiden IV (NTSC-U)
WarTech: Senko No Ronde (NTSC-U)

Games On Demand North American Store:
Akai Katana
Bullet Soul
Bullet Soul Infinite Burst
Caladrius
Deathsmiles NA
Deathsmiles IIX
Otomedius Excellent
Raiden Fighter Aces
Raiden IV
Shooting Love 200x
Under Defeat HD:DE

XBox Live Arcade:
1942 Joint Strike
Bangai-O HD
Galaga
Galaga Legions
Galaga Legions DX
Guwange
Ikaruga
Omega Five
R-Type Dimentions
Radiant Silvergun
Raystorm HD
Revolver 360
Sine Mora
Strania - The Stella Machina
KOF: Sky Stage
Trigger Heart Exelica
Trouble Witches Neo
Xevious

XBox Live Indie Games:
Aeternum
Aqua Kitty
Broken Pearl
Chronoblast
Game-Type
M.A.C.E.
Ninja STG
Pester
Platypus
Prismatic Solid
Score Rush
Shoot 1Up
SKH Resurrection
Snops Attack! Zombie Defense
Steam and Metal
Twin Tiger Shark
Valkyrius
Vampire Rage

Region Free/No IP Lock downloads:
(EU Marketplace)
Dodonpachi Resurrection
Dodonpachi Resurrection 1.51
Dodonpachi Daifukkatsu Black Label

(HK Marketplace)
Ginga Force

(JP Marketplace)
Dodonpachi Saidaioujou
Ginga Force
Mushi Mushi Pork & Pink Sweets
Shooting Love 10th Anniversary

Region free/IP Locked (must use a VPN to download to US Xbox 360)
Eschatos (HK Marketplace)
Espgaluda II Black Label (HK Marketplace)

Bonus Non Shmup Discs that work on NTSC-U xbox360:
Phantom Breaker Extra (J)
Cyber Troopers Virtual On Force (J)

So that leaves us with these REGION LOCKED disc releases from Japan:
DoDonPachi Daifukkatsu Black Labe, Cave 2011 (region lock japan)
Dodonpachi Daioujou Black Label X, Cave 2008 (region lock japan)
Instant Brain, Cave 2010 (region lock Japan) <Dodonpachi as an unlock able game>
Ketsui - Kizuna Jigoku Tachi X, Cave 2010 (region lock Japan)
Mamoru-kun Wa Norowarete Shimatta, G.Rev & Gulti 2009 (region lock japan)
Mushihimesama HD, Cave 2012 (region lock Japan)
Otomedius Gorgeous, Konami 2008 (region lock Japan)
Radirgy Noa, Milestone 2010 (region lock Japan)
Sakura Flamingo Archives, Milestone 2015 (region lock Japan)
Senko no Ronde Duo, G. Rev 2010 (region lock Japan)
Shikigami No Shiro III, Alfa System 2007 (region lock Japan)