So last year someone did this with the 3DS. It's called TWLoader and it's basically a DS flashcard for 3DS. It does just what the OP wants; it loads DS ROMs without a flashcard. It requires Arm9LoaderHax but it's worth it.
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Brand New
From Hong Kong
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Pre-Owned · Nintendo DS
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or Best Offer
From United StatesCustoms services and international tracking providedFree Returns
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Portable WiFi SDHC Adapter Reader Game Memory Card Flash Kit for 3DS NDS Device
Brand New
From Hong Kong
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Por WiFi SDHC Adapter Reader Game Memory Card Flash Kit for 3DS NDS Device Call
Brand New
From Hong Kong
Was: Previous Price$7.52
Free International Shipping
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Durable WiFi SDHC Adapter Reader Game Memory Card Flash Kit for 3DS NDS Device
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From Hong Kong
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Portable WiFi SDHC Adapter Reader Memory Card Flash Kit for 3DS NDS Device Strik
Open box
From Hong Kong
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Passcard the Perfect flash card boot up solution NDS & DS lite Genuine *
Pre-Owned
Top Rated Plus
or Best Offer
From United StatesOpen box
From Hong Kong
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Brand New
From Hong Kong
Was: Previous Price$7.52
Free International Shipping
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Supercard Mini SD Flash Card Adapter Cartridge 2GB for GBA SP GBM IDS NDS NDSL
Open box
Top Rated Plus
or Best Offer
From ChinaFree Returns
EZ Flash IV Flashcart for GBA and Nintendo DS (with MicroSD Card and Reader)
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Time left4d 18h left
23 bids
From United StatesCustoms services and international tracking providedSupercard Flash Card Mini Sd Card Adapter For Gba Sp Gbm Ids Nds-Ndsl HOT SALE
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From United Kingdom
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3 in 1 EZ Flash Omega for GBA/GBM/GBASP/NDS/NDSL Game Card Micro SD up to 128GB
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From China
Was: Previous Price$47.98
+$3.60 shipping
Mini Supercard SD Flash Card Adapter Compatible With GBA/ SP/ GBM/ IDS/ NDS/NDSL
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From China
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EZFlash 2nd Generation Game Cartridge EZ Flash Card For GBA GBASP NDS Console
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Pokémon Pokemon Jade Gameboy Advance Nintendo DS Keitai Denjuu Telefang 1 Hack
Pre-OwnedFrom Portugalor Best OfferNintendo 2DS XL Bundle - Black/Turquoise 30 pre installed games cfw hack
Pre-OwnedFrom United Statesor Best OfferNintendo New 3DS XL Launch Edition 32Gbwith hombrew lancher 2.0.0v hack
Pre-Owned · Nintendo 3DS · New Nintendo 3DS XL · 4GBTime left1d 22h left0 bidsFrom United StatesBuy It NowNew Nintendo 3DS XL (New Galaxy Style)
Pre-OwnedTime left6h 20m left0 bidsFrom United StatesCustoms services and international tracking providedNintendo 3DS XL bundle with 10 games and case
Pre-OwnedTime left10h 14m left1 bidFrom United StatesComplete Nintendo 3DS Aqua Blue Console. Adult Owned. Good Condition, With Game.
RefurbishedTime left7h 28m left0 bidsFrom Canada(HACKED) Nintendo *New* 3DS XL Launch Edition 4GB Black
Pre-Owned · Nintendo 3DS · New Nintendo 3DS XL · 4GBFrom United Statesor Best OfferCustoms services and international tracking providedHacked Nintendo New 3DS Animal Crossing Happy Home Designer Edition
Pre-Owned · Nintendo 3DS · New Nintendo 3DS - Original · 8GBTime left4d 8h left0 bidsFrom United StatesCustoms services and international tracking providedBuy It NowNintendo ds xl
Pre-Owned · Nintendo DSTime left7h 51m left7 bidsFrom United StatesNintendo 3DS Launch Edition Cobalt Blue Handheld System
Pre-Owned · Nintendo 3DS · 3DSTime left10h 36m left4 bidsFrom United StatesNintendo DS Lite USG-001 For Parts or Repair
Parts Only · Nintendo DSTime left13h 43m left0 bidsFrom United StatesCustoms services and international tracking providedC613 Nintendo 3DS console Ice White Japan x
Pre-Owned · 3DS - OriginalTime left18h 30m left2 bidsFrom JapanNintendo New 2DS XL Handheld System White and Orange
Pre-OwnedTime left11h 24m left19 bidsFrom United StatesC632 Nintendo new 3DS LL XL console Metallic Black Japan w/stylus pen case
Pre-Owned · Nintendo 3DS · New Nintendo 3DS XLTime left18h 14m left2 bidsFrom JapanHACKED Nintendo 2DS Super Mario Bros. 2 Console Bundle - Scarlet Red (READ DESC)
Brand New · 2DS - OriginalFrom United Statesor Best OfferNintendo 3DS console Blue x Black for parts Or Repair
Parts OnlyTime left10h 30m left3 bidsFrom United StatesCustoms services and international tracking providedC701 Nintendo 3DS console Cobalt Blue Japan
Pre-Owned · Nintendo 3DS · 3DS - OriginalTime left18h 19m left0 bidsFrom JapanNintendo 3DS XL [Custom DEDEDREAM Decal] - Blue/Black (SPR-S-USZ/F-C0)
Pre-Owned · Nintendo 3DS · 3DS XLTime left11h 49m left1 bidFrom United StatesBlue Nintendo 2DS
Parts Only · Nintendo 2DSTime left1d 18h left0 bidsFrom United StatesC696 Nintendo 3DS LL XL console Pokemon Center Xerneas Yveltal model Japan JUNK
Parts Only · Nintendo 3DS · 3DS XLTime left18h 20m left0 bidsFrom JapanNintendo 2DS
Brand New · Nintendo DSTime left5h 31m left8 bidsFrom United StatesNINTENDO DS LITE silver used condition is good & cool (A sticker is attached)
Pre-Owned · Nintendo DSTime left4h 42m left1 bidFrom JapanC554 Nintendo DS Lite console Gross Silver Japan NDS x
Pre-Owned · Nintendo DS · DS LiteTime left18h 32m left0 bidsFrom JapanSoft-modded Red Nintendo 2DS READ DESCRIPTION
Parts Only · Nintendo 3DS · 2DS - Original · 4GBTime left11h 27m left1 bidFrom United StatesNintendo DS Lite Polar White System with Charger and Stylus
Pre-OwnedTime left9h 49m left1 bidFrom United StatesCustoms services and international tracking providedC697 Nintendo 3DS LL XL console Animal Clossing Limited model Japan
Pre-Owned · Nintendo 3DS · 3DS XLTime left18h 14m left0 bidsFrom JapanNintendo 2ds pikachu Pokémon new
Brand NewTime left9h 34m left9 bidsFrom CanadaZ1141 Nintendo 3DS Light blue console Japan N3DS Junk For parts
Parts Only · Nintendo 3DSTime left6h 56m left2 bidsFrom JapanC706 Nintendo DS console Platinum Silver Japan NDS w/stylus pen
Pre-OwnedTime left18h 20m left0 bidsFrom JapanHacked Nintendo New 3DS XL Launch Edition Red Handheld System W/ 7 Games
Pre-Owned · Nintendo 3DS · New Nintendo 3DS XLFrom United Statesor Best OfferNintendo DS Lite Guitar Hero Silver On Tour Handheld System Excellent condition
Pre-Owned · Nintendo DS · DS LiteTime left11h 44m left0 bidsFrom United StatesCustoms services and international tracking providedC542 Nintendo DS Lite console Ice Blue Japan NDS x
Pre-Owned · Nintendo DS · DS LiteTime left18h 28m left0 bidsFrom JapanNintendo Ds
Pre-Owned · Nintendo DSTime left14h 38m left1 bidFrom United StatesCustoms services and international tracking providedNintendo 2DS XL Pikachu Edition Console
Pre-OwnedTime left18h 27m left0 bidsFrom United StatesC693 Nintendo 3DS LL XL console Red x Black Japan JUNK for parts
Parts Only · Nintendo 3DS · 3DS XLTime left18h 20m left1 bidFrom JapanNintendo 3DS XL Black Handheld System Bundle: Game, Case, & Charger (Smash Bros)
Pre-OwnedTime left15h 57m left13 bidsFrom United StatesCustoms services and international tracking providedNintendo DS Lite Metallic Rose Handheld System
Pre-Owned · Nintendo DS · DS LiteTime left6h 57m left2 bidsFrom United StatesC543 Nintendo DS Lite console Ice Blue Japan NDS JUNK For Parts
Parts Only · Nintendo DS · DS LiteTime left18h 32m left0 bidsFrom JapanNintendo DS Lite Console Polar White Bundled Lot: Video Game+ Travel Case Go Set
Pre-Owned · Nintendo DS · DS LiteTime left8h 54m left0 bidsFrom United StatesCustoms services and international tracking providedNintendo DS Lite Handheld Console Blue
Pre-OwnedTime left15h 21m left1 bidFrom United StatesC616 Nintendo 3DS console Misty Pink Japan x
Pre-Owned · Nintendo 3DS · 3DS - OriginalTime left18h 28m left1 bidFrom Japan