Pro Mini (which OP said they have) is listed as 5-12V input for the 5V versions (like I use myself).
The Mini (those designations get more and more confusing) is indeed a bit different again, but does have 5V pins.
SomethingClever:
It says here that the input power is 3.35-12v, so I guess the Feather's 3.3v is not enough.
I noticed that; it's indeed outright weird. I don't know why they ask for that extra 0.05V - nor do I believe it will make a difference. You can of course just connect it to the same power supply you use for the Feather, instead of taxing the Feather's built-in regulator.
The main problem is the overall power supply. An SD card can draw quite some power when active, and that's not what those little regulators can handle.