Most sponsorship in karting, at least at the local levels, is more a nice guy type thing. Someone is a friend of the family, or works at a place, or something like that. Just throwing some money or product at you because they want to be part of the “team”. The value they are getting for their money is in the spirit of giving. I agree that a website is probably a waste of time and money, particularly because social media has all but replaced them for this kind of thing.
As drivers get more experience and travel around to where lots of eyes are seeing them and their sponsors, then there is a possibility that a sponsor could realize some actual monetary value out of their deal with your driver, but this is still pretty uncommon in karting.
I don’t want to sound negative about getting sponsors in karting, but its important to be realistic and understand that very few of these deals are really benefitting the sponsoring businesses, other than making them feel like they are part of something unique. If your approach gets you money by filling their need to be part of something, then go for it! If people want to give you money to hang out with you, by all means, let them.
The real sponsorship in karting is getting free stuff from kart companies that want you to use their products. I would guess that 90% of the “sponsorship” in karting is this type of deal. Very few people, in the overall scheme, are actually getting cash to put toward racing, unless its done by Dad’s friend, or uncle Ron’s biz, or something like that.
Good luck and have fun!
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