Your correct in that there isn't any non-free firmware required for that
chipset. Even when a particular chipset is supported it does not mean that
all wifi adapters with that chipset are supported. Not all RTL8187X adapters
will necessarily work or work well. I know there are a number of different
adapters on the market which I would not recommend for one reason or another
even though they have free software friendly chipsets.
That said if this is a laptop you might want to check if it has an on/off
switch. Sometimes this is also an issue an its unrelated to the card itself-
so your card may in fact work after all (either on the system you have or
another). Sometimes nothing will work on the system if the switch is
permanently in an off state and can't be turned on. Special drivers are
needed for the machine in that case and those almost certainly don't exist or
they'd have been included in the kernel already.
In any event there is usually an FN key you can try (find your FN key and
then find the little wifi symbol on one of the F1-F12 keys). There are also
commands you can try running. Ultimately the best thing to do is a buy a
laptop / desktop with free software support from the start. If you don't at
least avoid the models with digital restrictions of the mini pcie card slot
so you can swap out the wifi card. Dell, HP, Toshiba, Lenovo/IBM, Sony, and
Apple at a minimum should be avoided.
Try this guide to see if it'll fix your problem:
https://www.thinkpenguin.com/gnu-linux/penguin-mini-pcie-and-half-height-cards
