From looking at the schematics, I'm going to have to say no. While I'm not 100% sure, The schematics show the USB 5v line actually only powering the FTDI serial conversion chip, and not going off to the connectors. If you really wanted to and had some expert soldering skills with SMT, you could solder a hookup wire between the necessary connectors, but this *could* (but not likely) create issues with powering the Tinyduino because you'd need a current limiter or else you'll probably find the USB device, depending on what it is, drawing too much power.
Also, assuming you just want to power the device, (refer to USB & ICP Shield schematic PDF, available on the site) adding in a hookup wire between the USB connector's pin 1 and the 8th pin on the connector (which one is number 8 I don't know) should do the trick, provided you use a voltage booster to bring the LiPo's output up to 5v.