Ubuntu is based on Debian; usually what works on one will also work on the other. I really think it has the drivers out-of-the-box.
Maybe linux-hardware.org can help you somehow?
Part 1, part 2, probe 1, probe 2, probe 3
This is a very interesting option… So I tried to follow the steps – and this is what I get (see image). The WIFI card in the ThinClient - is a Broad
I was for-sure thinking this was the 'fix" - 
OK – I figured out the error here – I had a misspelled aports (there was an ‘i’ in there).
So I went back through the process - but it fails as it states that I am logged in as ‘root’ – cannot figure out how to login as another user. I tried the ‘su - {usernamme}’ screen filled up with … (I had to reboot to get out of this).
Did you try to run the commands with sudo?
It doers not like that… It appears that the TERMINAL – you are in the DOCKER.
In Docker… you are ‘root’. Every time that I try and change to another user – system freaks out.
Maybe try to use “su -c” instead?
Might be an idea … if I knew what that command was, and how to use it.
I downloaded DEBIAN 11 - made a bootable USB - now just gotta swap out the SSD with another - and install.
I honestly cannot believe that this process is THIS HARD.
Where are the HASS.IO DEVELOPERS??? I mean come on folks - not everyone wants to have an Ethernet Cable plugged in all the time - especially, when central home placement is key…not everyone has an Ethernet port in the middle of their home. 
Curtis
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These are the steps I am following:
https://wiki.alpinelinux.org/wiki/Wi-Fi
All is good until I get here (as this wireless card is a Broadcom card):
Change your directory to aports/non-free/b43-firmware, then build it.
Tip: You can’t be root and must be a user of the group abuild (use groupadd f.e. addgroup $(whoami) abuild)
Tip: If this is your first time building a package you will need to generate a key for use in signing packages (use abuild-keygen -a -i)
After much playing and learning (and getting no-where with any help from the ALPINE forums) – I have two things to report.
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I found that testing other miniPCI wifi cards – the INTEL wifi cards are recognized out of the box in Alpine Linux (I installed ver. 3.15 on an old Thinkpad - everything was recognized - except the Bluetooth USB dongle I had plugged in – but I had problems with it in Ubuntu too). Since Hass.io uses Alpine Linux 3.15 - I have ordered me an INTEL miniPCI wifi card to go into the HP t620 ThinClient box - then gonna test the Intel NUC install there - and see what happens.
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Right now on the HP t620 ThinClient (which has a Broadcom miniPCI wifi & bluetooth card that came with it installed)…I have installed Debian 11 (no GUI) and have gotten both wifi and bluetooth working on it. Then installed the HA Supervised (GitHub - home-assistant/supervised-installer: Installer for a generic Linux system) - and that is also running. I used these steps Installing Home Assistant Supervised on Debian 11 Bullseye and had to do a few tweaks which are discussed within the posting (namely I found that installing things (even Wired) does not like IPv6) - once I disabled that - installation completed.
So, as of this posting - I am running on Debian 11, the latest HA - no errors, and no Ethernet cable (yeup - wifi).-- Waiting on the INTEL wifi card to get here - then I am going to build the system which hopefully will remain in place…while I wait - I am gonna play with the ZWave and some other things.
Curtis
! UPDATE !
Intel Wireless Card (Centrino 6235) arrived last night. I have taken out the Broadcom card which was in this HP t620 and installed. I had to update the firmware/drivers for the Debian 11 to see and use the card. Once that was done – AGAIN, I have wifi in Debian 11 (now with an Intel Card).
I will do the INTEL NUC installation later - and find out if am able to get wifi in a ‘pure Hass.io’ installation using the HomeAssistant build.
Curtis
Did you manage to get this working? Got the same problem with my t630 not seeing the wifi internal build in
I did not - not on the HP T620. I just took out the WiFi card (with the intent of putting a CORAL TPU in there at some time) and I temporarily used a USB WiFi and gave that up too. Just using the Ethernet port.
Just install image for NUK in proxmox, will works with Your pcie
Is there someone who got this to work in the meantime?
Got the same T620/Broadcom issue.
External USB dongle does work. But I want it internal 
Switching to HA Supervised installation is the best solution, even though there are more maintenance of the Debian host OS put into the administrator of the HA installation.
But if you have the skills to try to install a WiFi driver into a stripped down Alpine Linux, then maintaining a Debian 11 is probably not a hard task.
I opened an issue on home-assistant/operating-system for my card and somebody added the driver into the next official release. Maybe you could do the same.
its not working with nuc image
The wifi card in my HP Thinclient T520 worked fine on the stock windows. But doesn’t show up on HassOS.
I tried some other ideas but eventually did the same as andrei and submitted an issue for my Broadcom BCM43228 802.11a/b/g/n Wifi card which should work in Ubuntu.
Will let you know if it works!
The issue is that the drivers for some netcards, like broadcom and intel, are released under a license that makes it impossible to include it under many of the licenses that many Linux distributions come with.