I think you have it right there in the difference between what you quoted - the Lord's Prayer doesn't say "do not tempt us" (and James agrees as to why) and James does not say "God does not allow people to be dragged away and enticed" (which would make the world a very different place).
A prominent example of God explicitly allowing someone to be tempted is the book of Job.
And the LORD said to Satan, “Have you considered my servant Job, that there is none like him on the earth, a blameless and upright man, who fears God and turns away from evil?” Then Satan answered the LORD and said, “Does Job fear God for no reason? Have you not put a hedge around him and his house and all that he has, on every side? You have blessed the work of his hands, and his possessions have increased in the land. But stretch out your hand and touch all that he has, and he will curse you to your face.” And the LORD said to Satan, “Behold, all that he has is in your hand. Only against him do not stretch out your hand.” So Satan went out from the presence of the LORD. (Job 1:8-12, ESV quoted)
"Lead us not into temptation" would be a prayer specifically against that sort of scenario. Eve's being enticed by the serpent to partake of the fruit of the tree of knowledge of good and evil (Genesis 3) might be another example. Being led into temptation also happened to Jesus himself, according to Matthew 4:1, just two chapters before the Lord's Prayer:
Then Jesus was led up by the Spirit into the wilderness to be tempted by the devil.
In all these cases it is not God doing the tempting - the devil, the serpent, Satan are specifically named. God may indeed lead us into temptation, but James points out that God is not the tempter—likewise we might say, the trainer that brings us into the arena to fight is not the opponent we face.