What runs though the board and what doesn't? If you want to not record the house, then everything will need a mic.
Ideally, you would have the recorder on an aux so that you can fine tune the recoding mix as the house mix may not be ideal. Or, you would use direct outs or a split to a multitrack.
I had decent results at a show over the holidays where there was a Rode NT-4 out in the pews (it was in my favorite seat) recording to two tracks, and the vocals (I think they were post EQ but pre EFX) going to another two tracks. Not everything was miced, so this was the only way to record everything. I didn't do the final mix, but the end result was good.
Also keep in mind that you may not necessarily have rights to reproduce anything that you record. Performance of copyrighted material in worship has a special exception(?) in the US and Canada, and CCLI only covers reproducing the lyrics. CDs, MP3s, etc, would require a mechanical license from the publisher.