I just realized that I'm down to only 1 of the Sony HD tapes, and nowhere around here carries them. I'm going to need them for tomorrow but doesn't look like I'll be able to get them. Am I still able to shoot in HDV (I have the HC1) with a regular minidv tape?

Yes absolutely.
Any MiniDv tape wil record HDV video.

Recommend due to possible lubrication-compatability issues that it's preferable to stay with one brand, so if you can get hold of Sony Premiums for example they should be fine.

I use Sony Premiums 95% of the time and are fine. The very few times i've had dropouts are when i ran out of sony Premium and used a JVC tape. Those tapes caused noticeably more issues than Sony tapes, so it's my personal opinion not to recommend JVC tapes, although others mileage may vary, as they say..

Thank you very much!