Thomas R. answered 06/24/19
Over 25 years of experience and a sense of humor about math
Technically, the two expressions are different. "Can I?" means you are asking if you have the capability to do a thing. When students ask me, "Can I go to the bathroom?" I sometimes joke, "That's between you and your doctor." "May I" is a request for permission. Prescriptivists strongly endorse a rules-based system where you shall suffer mightily if you end a sentence a preposition with. Never split infinitives! Commit no comma splices!
Other language mavens feel that language is an evolving thing, and that rules should serve only to guide and clarify. Did your English teachers tell you that we are meant to avoid sentences with final prepositions on the word of one British dude a few centuries ago who suddenly declared this a Very Important Rule? One reason that the rules matter so greatly is that English is literally a bastard language. It is cobbled together from Norman French, Latin, a couple of words of Russian and one of Polish, plenty of German, and some Spanish words if you live in America. Vowels were added and went silent thanks to the Great Vowel Shift, which quieted certain e's at the ends of words. Earlier, we had silenced the leading "K" we had inherited from Viking invaders. This is why pronunciation is such a muddle in English.
All of this is by way of saying you should follow the rules in your formal correspondence,if only to show the recipient that you know The King's English. In casual setting, relax and say whatever -- but be prepared for someone like me to tease mercilessly if you ask, "Can I eat a hot dog?" That's between you and your dietitian...