Should I shoot at 1080p if I intend to use 720p on the web?
There is no one right answer to this very general question, but here are the considerations.  Some cameras (such as the Canon EOS DSLRs) have very naive line-skipping algorithms to decimate their normally very high resolution down to FullHD or HD video.  Such algorithms create lots of aliasing noise, which is further aggravated by further image scaling.  Other cameras are more intelligent about how they match the sensor's native resolution to the desired recording resolution.  RED cameras, for example, crop down the sensor so that whether you are recording at 6K or 5K or 4K or 3K or 2K HD resolution (1920x1080) you are using precisely the RAW pixels, not some interpolation or decimation thereof.

Another consideration relates to the actual recording bandwidth of your system.  If you are recording at a fixed rate (such as 28Mbps AVCHD), then a 1080p video will, necessarily, have to employ greater compression than the 720p video, because the 1080p has 2x the spatial resolution of a 720p video (and both have the same bit rate).  If your fixed recording rate is 200Mbps (such as is offered by the Panasonic GH4), that might be so generous that you really are getting very high quality for both 720p and 1080p.  So it depends on whether your camera's imaging system is well within, at the limit of, or past the point of a good compression tradeoffs.

Finally, 1080p streaming might strip out way more quality from your 1080p master than exists in even your 720p master.  In which case a high-quality 720p master might be all you need.

The places where video quality really gets down to the pixel level are typically within the post-production process, pulling keys, doing color correction, sharpening, etc.  Once you have a high-quality master, streaming compression will probably obliterate all differences in your 1080p vs. 720p masters.  Unless you have a notorious camera known for particularly poor quality in the first place.