When you want to take part in some activity and you are late, you can say either that you are “late for the activity” or that you are “late to the activity”, for instance,
He is going to be late for/to work.
Don’t be late for/to dinner.
However, the variant with “for” is much more widespread than the variant with “to”, and many native speakers (especially of British English) consider the phrase “to be late to something” either unnatural or substandard (although both are relatively common in English literature). If you want to be on the safe side, it is advisable to avoid the variant with “to” altogether.
One major exception to the rule is the colloquial phrase “to be late to the party”, which means “to become involved in something long after others”. For example, you could say:
Again, this idiom is less widespread in British English than in American English, but in American English, it is actually more common than “to be late for the party”. Since it is colloquial, you should avoid it in any kind of formal writing.