Its one of those things that they don't encourage or promote, but as long as you are not making it too obvious ie large coolers/ picnic hampers, then I don't think they are likely to stop you.
They don't provide picnic areas as far as I'm aware, so they are really not keen on you doing this for obvious reasons.
It would be very uncharacteristic, for any Disney employees at least, to 'get nasty' with guests...its not the sort of tone they want to set.