It turns out that this 16MHz machine is quite capable, as I managed to connect it to wireless internet and post a comment on this very page:

I used a tiny A5-V11 router with RT5350F SoC: https://openwrt.org/toh/unbranded/a5-v11 that I flashed with OpenWRT. Currently I am connecting it externally to the ISA ethernet card attached to T3200SXC, however I plan to embed this tiny OpenWRT module directly onto the ISA card. This is how it would look like piggybacked:

It uses just around 1-2 watts of power, so I plan to find a suitable soldering point with +5V on the wireless card and solder wires in series with a small polymer fuse. The ethernet wires will need to be soldered between these two cards - i may desolder the RJ45 connectors completely. To avoid blocking the wireless signal, metal bracked will need to be replaced with a 3D-printed one.

On the software side, I prepared some scripts for both the MS-DOS machine and OpenWRT router, that allow for very simple scanning and connecting to the networks directly from MS-DOS. Details will be stored in this repository: https://github.com/adbrt/dos-wifi-solutions/

For a web browser, I'm using Links for DOS: http://links.twibright.com/download.php

It supports TLS 1.2 and it even has a graphical mode! However it slows things down quite a bit:

I also managed to connect to a Wi-Fi control interface of a digital camera (Lumix GH5). The camera detected that the connection has been estabilished. I wonder if I could take some photos, download them to the DOS machine and upload them online using HTTP interface of the camera. I will need to look into the protocol and try some things out. You can see here how the wifi scan / connect command works:

By the way, as a bonus, I will mention that old-school ISA 2.4GHz wireless networks adapters did exist, and in fact here is one that I tried to use with that computer!

However it is completely incompatible with modern Wi-Fi protocol - so it wouldn't be much of an use. Therefore I'm going with the embedded OpenWRT module option.