There is no firmware or anything on anything with a GBA slot that makes for a reasonable means of running GBA code like you might be used to with later systems. With a DSi or 3ds you could use emulation that gets basically there but eh, same for a DS with an enhanced flash cart (the baseline DS is not really able to sensibly emulate a GBA, the DSTwo family being the main thing that can owing to it having onboard extra hardware to do such things but you might find an ISMM somewhere).
Two broad options to do things in hardware* (emulation has long existed, indeed other than save types a current emulator at the time of the NA release will probably do most of the library just fine) then are
1) Multiboot cable.
2) Flash cart.
*this is ignoring the likes of the analogue pocket which I would argue is hardware but includes options to run from its own internal loaders.
Multiboot cables are limited to about 256KB of data and naturally need a device (basically going to be a PC, there are some things for gamecube and standalone devices but don't).
I am not aware of any being sold these days so build your own
http://problemkaputt.de/gbatek.htm#auxxboopctogbamultibootcable
https://web.archive.org/web/20150810222526/http://reinerziegler.de/GBA/gba.htm#Multiboot
https://web.archive.org/web/20150810170955/http://www.devrs.com/gba/files/mbv2faqs.php
The main bonuses to them are theoretically quicker if you are testing out code (though eject SD card, shove into reader, copy files is plenty quick) and ability to have your GBA speak to your PC (said code might take in info/data from the PC beyond that) which for 99% of people means you can use your GBA as a PC controller if for some reason you want to do that (they are nice to hold but why I would when I have any number of other controllers I don't know, some like to emulate GBA on PC and control as though they were though).
Flash carts...
They have existed for a very long time (possibly longer than emulation but it is months at best) and other than the very first ones and supercards (and clones thereof, cyclops doing one) then if it fits on the cart itself it will run it bug for bug compatible with all but about 9 ROMs, 7 of which have patches that do most things for most people,
https://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995 for the list of trouble games and fixes.
Like with most things you can categorise them in any number of ways but I will attempt one here.
1) Very oldest stuff. If you ever see speed patches and such it is probably for these (or supercards but more on those later). Avoid and frankly you will be hard pressed to find it.
2) NOR era. The GBA read speed for ROM data is actually really quite high for the time and necessitates fast memory to run it. This might even trouble SD cards (which are basically NAND memory) today, certainly did at the time. The main solution other than RAM (more on that next) which has the problem of being small, expensive and needing power to keep its data is NOR memory, which is expensive per megabyte, has some quirks that limit useful maximum sizes and the like.
These will tend to come with either cradles/linkers in which to put the carts to write to/read from (which is also a nice way to dump GBA games) or custom cables that plug into the cart or are the special twist on the multiboot stuff above, probably going to need a Windows XP machine (or virtual machine) to sort them. Some limited options for some carts if you have a DS/DS lite and DS flash cart to write things.
3) NAND era.
Rather later in the GBA lifetime (the DS would hit not so very long after) there came these.
Rather than trying to add more and more NOR they opted for RAM and a big chunk of NAND.
The EZFlash 3 (different to the 3 in 1 we cover shortly) being the main reference point for most, though the G6 also features here.
4) DS GBA slot era.
Overlaps a bit with 3) actually but rather than sticking a bank of NAND on the cart itself things moved to using CF and then SD cards instead (lets the user pay for it, lets the user swap out when they like, and swap when it breaks as it is common enough to be annoying). The DS was also a thing by this point, and hacked, so people were trying to fit the far larger DS games on legacy GBA era things and thus people wanted more size.
This is where Supercards, M3 and the EZFlash 4 came to the fore. Supercard sacrificing GBA abilities considerably, the others having models that diminished GBA options in favour of being a cheap DS player. You may also encounter the G6 which is more like 3) but of the 4) era. Their main focus for most of this was DS games so many will lack finely honed features some of the 2) and 3) stuff might have like cheats, savestates and what have you.
5) Expansion pack era. Might realistically be 4a) but oh well.
Flash cart teams realised that if you had a DS slot flash cart you probably had all the storage you needed (average GBA game is 8 to 16 megabytes, SDHC 8 gig or 16 gig cards were reasonably affordable at this point). Consequently they went back somewhat to the NOR era but also NAND with small carts and say 256Mbit/32 megabytes of NOR (the maximum for most purposes of GBA ROM size) and were built for DS cards to manage them. If you can make them fit (the DS lite features a reduced size GBA slot and was popular at the time hence most things for it being that size) then they will work on a GBA but it is not worth it outside of specialist things like faking a GBA game for the purposes of DS games that see their older titles and unlock bonuses accordingly
https://nintendo.fandom.com/wiki/List_of_Nintendo_DS_games_with_GBA_connectivity . For use with pokemon for such things you tend to want a DS flash cart as well
https://gbatemp.net/download/pokepatch.27240/ . Options to do this with others here are available depending upon what it is and might need some effort, for most purposes get the save, load it in an emulator and fake it with that.
6) Modern era.
For many years (see the link above on trouble games, the thread was made when such things were getting scarce).
Believe it or not other than the GBA and things following it then flash carts for most other consoles was limited to legacy old nasty things (as in would have been the height of tech in the same years the consoles were current and not much later), and trying to twist whatever possibly deliberately limited homebrew flash cart was made to working for your purposes. This was until everdrive came along and rocked that whole world with their stuff which brought it kicking and screaming into the modern world.
They tried the same for the GBA but as mentioned above all but the oldest and weakest will play anything in the GBA library in a bug for bug/as it was on original hardware (if not better) manner. Still a very nice GBA flash cart from them though, quite pricey but justified for many. EZFlash also fired back with the Omega line (which also saw a further successor with the truly amazing Omega definitive edition).
You will pay up here but they are basically everything you probably could want from a GBA flash cart. Can carry basically the entire GBA library, homebrew to a point (music and video players exist, though nothing amazing or all that useful compared to basically anything else you might have running today, so obviously your choices are infinite there. All the things the GBA can emulate though
https://gbatemp.net/threads/flubbas...s-plus-assorted-other-recent-homebrew.619739/ http://nintendo-ds.dcemu.co.uk/gba-emulators-1158176.html you could probably crowbar in there as well) and ROM hack on you if you really wanted, certainly several hundred of your favourite games is nothing to them where if you go back to 2) above then you might get 6 on the average one if you choose carefully.
Short version of that. Ignore multiboot unless you know it is what you want (you didn't ask about/mention any coding you were intending so you don't want it).
There might be some old EZFlash 4, reform or redux carts out there in some vendor. They won't have fancy options like savestates and onboard cheat support is spotty (easy enough to hardcode cheats in
https://gbatemp.net/threads/gbaatm-rebirth-gba-auto-trainer-maker-a-new-hope.564321/ ) but will be cheaper and allow you to play basically everything with minimal fuss as far as I am concerned. Avoid any GBA slot supercards and the limited GBA option things of other makers (there is a reason you might still find them in some shops all these years on -- nobody wants one).
If you have a DS and DS flash cart there are probably some EZ 3 in 1s around, probably need a battery replacing but easy enough if you can solder. Theoretically you could use it with a GBA but... don't.
If you are willing to spend a bit then the modern offerings in flash cart world are pretty nice. See
https://gbatemp.net/review/everdrive-gba-x5-mini.1575/
https://gbatemp.net/threads/ezflash...on-gba-flash-cart-in-house-at-gbatemp.581991/
https://gbatemp.net/review/ez-flash-omega.765/
https://gbatemp.net/review/ez-flash-omega.795/