I would suggest not starting in the 2.4GHz band, the protocols used there (wifi, bluetooth) are very complicated to understand. Get a rtl-sdr and start with something simpler: FM broadcast (my blogpost: https://www.abclinuxu.cz/blog/jenda/2019/11/gnu-radio-first-..., there is even an example capture you can download and replay, so you can start without the physical radio), police radio (both analog and digital), radiosondes, weather satellites, ISM stuff - temperature sensors, garage and car remote controls, airplane multilateration (https://github.com/mutability/mlat-server )… You can do lot of stuff even with the $10 rtl-sdr, for example I have used it for multilaterating TV and radio transmitters (thesis: https://jenda.hrach.eu/dipl.pdf, unfortunately "layman's explanation" is available only as a lecture in Czech). I know people are even building radioastronomy stuff and passive radars (https://www.rtl-sdr.com/passive-radar-dual-coherent-channel-...) with rtl-sdr.
Once you have basic understanding of the topic, you can get better hardware: AirSpy (the same features as rtl-sdr, but MUCH better signal-to-noise ratio and bandwidth) or bladeRF (costly, but probably the best radio you can get now). For example I'm now building a weather radar based on bladeRF. The bladeRF has a FPGA with open-source HDL, so you can mess even with absolutely lowlevel and bleeding edge stuff.
Going back to your original question:
Most cards load firmware from a file when they are initializing (check "dmesg|grep firmware", on my machine, for example, it says it has loaded /lib/firmware/rtl_nic/rtl8153b-2.fw), you are free to modify it. However, all (or maybe almost all) wifi cards have the format of the blob completely undocumented so it would be very hard to make a modification that would allow you to transmit/receive arbitrary signals. Something similar has been achieved with GSM phones (see OsmocomBB), but it requires very complicated reverse-engineering.
Firmware is typically loaded directly onto the hardware, so you would have to have a way to flash your custom firmware to the wifi antenna/card.
In the US and other countries, it is to the best of my knowledge legal to modify firmware for hardware you own. The illegal part is broadcasting, most bandwidths are highly regulated. Listening on the other hand is mostly legal, or at the very least extremely likely to fly under the radar.
You will not be able to modify the firmware blob just by running linux.
Few years back there was danger of FCC de facto banning alternative router firmwares like openwrt to prevent tampering with the wifi cards firmware.
This was the "only manufacturer signed firmware allowed" thing that thankfully was avoided.
FCC's motivation in this is to prevent people from using too much power or certain frequencies.
And as most manufacturers want to be able to sell in USA, it would have likely affected all versions.
Kainda like how many wifi devices sold in Europe only go up to channel 11 on 2.4GHz, when the EU band goes up to channel 13.
But ch12 and ch13 are not legal in usa, so they are blocked.
