I run an ESP8266 module from the 3.3v rail. It averages 80mA but has tiny peaks of 400mA (less than a microsecond).
In practice, it runs fine. If I remove and replace the module while the LE is running, the LE will sometimes reboot. That makes me think that I could be on the edge of the power supply's ability to keep up with the demand (at 3.3v). Adding a decent size cap would probably prevent that.
You might want to look at my post about it. I designed a tiny PCB with no components needed (just header pins). Its simple and extremely inexpensive, and has a large install base in a slightly different application.
