KirAsh4:
By the way, measuring voltages I get a steady 5.09V on the high side, and 3.294V on the low side. They don't budge when I try to access the card.
Can you try a different card and see if it works?
Different SD cards need different voltages. A lot of cards want 3.6V, yours might be one of them.
You're supposed to query the card and ask it how many volts it wants when you connect to it but the SD card library is a bit loose on that part of the protocol (IIRC). It assume SD card shields will all have 3.6V. If that's the problem you'll have to boost your voltage a bit...