Did Tanjiro use the lighting breath?
Yes he uses the proper breathing technique of lightning and it seems like you're asking if Tanjiro can use the forms of the breath of the lightning, Breathing technique and the forms of the technique is (2) different things so the answer is No.

Learning the proper breathing of the technique doesn't mean you will be able to wield your sword just like the other breath of the lightning users it requires proper training and talent to achieve such thing because by learning the proper breathing you'd just allowed yourself to pinpoint the parts of your body to enhance and perform the forms of the technique properly.