Have I correctly used 'successor' in the sentence?

I think it's correct..

No, that doesn't sound very good. You are usually a successor to someone or to some function (e.g. office, throne, etc.), certainly not to a business - I wouldn't say it that way. You could say:

The young prince is the successor to the throne.

If I rewrite the sentence as follows, will it be acceptable?

His parents raised him to be the inheritor/heir of their business.

I wouldn't use "heir" (it's "heir to", by the way) either . . . "inheritor" . . . is better, although I still wouldn't frame it that way.

I've just thought of another reason why "successor" is a bad idea. A successor is someone who succeeds - if he is the successor to the business, it means the business was replaced by him, which is definitely not the idea here.

XerxesNo, that doesn't sound very good. You are usually a successor to someone or to some function (e.g. office, throne, etc.), certainly not to a business - I wouldn't say it that way. You could say:The young prince is the successor to the throne.

Ok, Xerxes. Thank you.