Generally speaking to properly identify your devices I'd suggest to execute
lspci -nnvv and
lsusb -v, but those tools require the installation of the 'pci-utils.tcz' and 'usb-utils.tcz' extensions for them to be run (which could leave you in a catch-22, if the wireless USB adaptor is your only network connection). As those tools should be present on pretty much any other distribution (if they are more "bloated" than Core) I would not see an issue to run said commands using a different Linux (of course provided you've connected the exact same set of devices you plan to use under TC). In any case please attach the output here, if you want us to take a look at it.
Generally speaking you've got at least three principle options for an X server in Core:
(1) the default 'Xvesa' which is included in the TinyCore ISO,
(2) one using frame buffer via the 'Xfbdev.tcz' extension, and
(3) the 'Xorg' one for which the TC 4.x repository offers different versions via the 'Xorg-7.4.tcz', 'Xorg-7.5.tcz', or 'Xorg-7.6.tcz' extensions.
Now, your easiest option might be to boot TC and just run
Xvesa -listmodes. If that output contains an option you like (e.g. '1024x768x16') try to reboot TC again, but this time use the 'xvesa=...' boot code (e.g. 'xvesa=1024x768x16') at the respective prompt.
Likewise, to find you which frame buffer modes might be supported by your hardware you could boot with the 'vga=ask' boot code and follow the respective prompts. Mind you any 'vga=...' boot code has no influence on your 'Xvesa' server, as it still requires the 'Xfbdev.tcz' extension. I personally don't use it, but your situation might be different.
Finally, the more "full blown" Xorg server (which pretty much all the more "bloated cousins" are using) should support pretty much any VGA hardware, albeit for the "price" of all the extensions that need to be downloaded once and installed in such a way that they persist.
Generally speaking, if your WiFi USB adaptor is supported by another Linux distributions the chances are pretty high that also TC will be able to support it. Alternatively to the proper identification process as outlined above you could download the
CorePlus ISO and try that instead. It should contain all the extensions (and some more) that one requires to support the various WiFi adaptors.