I have a Raspberry Pi 3 w/ an Intel Movidius Neural Network Compute chip attached via USB. It has recently come to my attention that I may be asking the Movidius chip to perform too much work and as a consequence a thermal fail safe is being tripped in it after excessive use at room temperature. I want the whole thing to be contained within an enclosure for an outdoor application. This thermal problem has me worried since if the device is in a metal box for security/safety there won't be any air circulation to cool it down. Adding sunshine into the mix will cause the whole thing to heat up real fast.
Yeah, problematic, but not too bad, especially if the box is mounted in a stable orientation. If that's the case, you can put a hat on the box to shield from the sun. In addition the box will have to have an adequate ability to transfer heat to the outside. This can be done by using the box as a heat sink, which may involve a fan inside the box to circulate air, and requires an adequately sized box for the load. You can improve this by using a thermally conductive box and ensuring that ambient airflow and/or convection will circulate air over it. If the box will be too large, it may be an option to have heat sinks that actually protrude from the box and or heat sink the box itself. Depending on the situation with drainage and humidity, you may be able to filter ambient air and draw it in through the bottom of the box and exhaust it from the top under the sun hat, so as to produce an air curtain effect. An external fan and/or radiator with liquid cooling can be used for large loads.
Would integrating a Peltier cooler into the mix so that the cooler is in contact with the Movidius chip on one side and the interior, metal wall of the enclosure is on the other be a good usage of the cooler?
This is an example of using the case as a heat sink. If you were to do this, it may be better to cut a hole in the case and mount a heat sink over the hole, sealed, and then mount the peltier to the inside of this heat sink, bypassing whatever paint and/or potentially less-thermally conductive metals are in the case. However, the point of a peltier is to cool something aggressively and/or below ambient temperature, and as indicated in @evildemonic's answer, it actually necessitates a stronger cooling apparatus because of the addition of it's own losses. It's losses are mitigated in situations where the device being cooled's properties are increased by the difference in temperature (such as an LED laser) by a degree greater than the losses of the peltier itself. So for your application, they won't be the best.
Should I be using a fan instead? Since the device is outdoors it has me worried that rain/bugs or other problems could crop up if an entrance is present into the core of the enclosure. I have no experience building electronics that are intended to function outdoors and don't want any surprises.
You do have to plan for whatever ambient problems exist. If you need to, you can use various types of heat exchanger to avoid cycling ambient air directly through your unit, or filters and/or physical isolation should you decide to do so.