Due to the two latest files I'm now reasonably confident that the following is correct:
(1) The device in question is recognized by Puppy as 'Texas Instruments ACX 111 54Mbps Wireless Interface [104c:9066]'. I wanted to be sure as this
page lists several different devices with a productID of 'HWC54G'. Mind you that was what I kind of expected as the 'lsmod' output contained a subtle "hint" in this direction. What is OTOH mildly confusing is that 'lspci' lists 'acx_pci' as the kernel driver in use, whilst 'lsmod' does not show a module of that name. BTW, which Puppy version are you using?
(2) Looking trough the TC 3.x kernel sources I was not sure whether TC 3.x would support your card. Furthermore searching our forum here (and reading some older posts from user 'MikeLockmoore' mentioning such a device) I've now come to the conclusion that TC 3.x might not do it "out of the box" (or even with the respective extensions installed). Initially I suspected that it might be a case of missing firmware, as the 'dmesg' output showed the need for such, but I guess that is less likely now.
You could give it another try with TC for example by installing the 'wifi.tcz' extension and attempting to configure your card. We expect this to be a failure, but it might be just enough for TC to try to activate the card and that could leave some helpful trace in the 'dmesg' output. BTW, I personally find it easier to redirect output (e.g. via
dmesg > dmesg.txt) and then attach that file here instead of doing copy-paste from screen output, but that it up to you.
Anyway, your best bet is IMO to hope that Mike reads this thread and for him to supply you with the guidance of how to overcome your issue. You could of course (politely) "poke" him a little bit via a direct PM.