As the mom of an active 15 month old, I second the duct tape idea. I should have video'd my attempt to get a portrait of him on his first birthday (he's been walking since 10 months). In the end, I got a few semi-decent shots by handing him a distraction - foil wrapped roll of 120 film. I was using my Pentax AF400T (not a small flash) shooting through an umbrella to spread it out a bit. I had to tone it down to not blow things out (smallish area - 10x10ish), but the biggest problem was keeping him in the area covered by the flash and by the camera. It really didn't come down to how much light there was - just whether or not he was even in the frame and facing even sorta in the right direction to be lit nicely. The better shots were later when he was strapped to the high chair and had a cupcake for distraction.
When they get a little bit older, they are much easier to keep still, and will even do so when asked most of the time.
I do a few shots of each of my now 4 year old and nearly 6 year old boys each birthday and the challenge becomes a little easier each year.