Can LED strips light a small light box for a point-and-shoot camera?

In the most basic sense, yes. The colour temperature of the LEDs is well within the range of normal daylight, and just about any camera can cope with that.

The problem is that it's highly unlikely that the LEDs you are planning to use have a good colour rendering index (CRI). All of the consumer LEDs I have seen have a CRI in the low 80s at best. That means that there are substantial portions of the spectrum that are missing and/or over-emphasized, and you won't be able to fix the colour problems in your pictures by adjusting either the colour temperature or tint settings for the photos. The light isn't there, so the data aren't recorded. There are LEDs with reasonably good CRIs (you need something in the 90s at least), but they tend to be painfully expensive.

If possible, fluorescent or incandescent lighting would be better from a bang-for-the-buck point of view. Incandescent lighting is great, but there can be some significant heat problems, especially in a confined space. (Remember the Easy-Bake oven?) Fluorescents are bulky and a bit more expensive, but you can usually find bulbs with good-enough colour rendition at something approaching reasonable prices.