Is it permissible for someone who wakes up in a state of major ritual impurity (junub) five minutes before the Fajr adhan to perform tayammum, make the adhan, and lead the people in prayer as an Imam? If not, what should he do?

If the Imam's performing tayammum is due to a valid Shar'i excuse, such as the absence of water or the inability to use it due to illness or cold with no means to heat the water, then his prayer is valid. However, if his tayammum is without a valid Shar'i excuse, then his prayer is invalid and he must repeat it. As for someone who prayed behind him while unaware of his condition, their prayer is valid.