Current wisdom suggests the two words can/may be used interchangeably, but "may" is considered a little more polite making it more appropriate for formal situations.
I agree with those that say that both can & may should be accepted. English allows both precisely because English has almost completely lost the distinction between indicative and subjunctive. One way to display distinction between can and may in Spanish grammatically is the distinction between the indicative and subjunctive. But is there a difference in this sentence between Spanish subjunctive and indicative? I don't think pueda works here, so I don't think Spanish would use subjunctive here, but we will need a native speaker to chime in to confirm. "Pueda llevar su bolsa?" If one wants to be polite and deferential here, then su is used. "Peudo llevar su bolsa." The pronoun tu indicates familiarity and deference, not subjunctive in this case. In colloquial English, "Can I carry your bag"--said to someone you know--is the norm in the midwest U.S., whereas "May I carry your bag" is more formal and would be said to someone you don't know--the granny who is having a hard time--and so su would be needed in Spanish equivalent. IMO, the use of pronoun tu here indicates "can" is best but I wouldn't fall on my sword over it.
Note that "puedo llevar" is a verb expression that means "I can take/carry" (or in question form, "can I take/carry"). In Spanish, you never conjugate the second verb in such verbals. This is different from English, where the second verb following a modal like "can," "should," etc. is conjugated to second person singular. Spanish never does that, even though English sometimes does and sometimes doesn't. For example:
"I must go" = "debo ir"
"I need to go" = "necesito ir"
"I should go" = "debería ir"
Quite often, Spanish requires an intervening preposition, where English does not. For example:
"I have to go" = "tengo que ir"
"I'm going to go" = "voy a ir"
This is just one of those cases where English grammar doesn't help with the Spanish grammar.