Think of "any" as meaning "some, no matter how much or how little". "Some" is more appropriate when you mean "a certain amount of", though you are not trying to specify what that amount might be. In this case I would say that you use "any" because the "if" condition is satisfied even if you only need the tiniest amount of information. We follow this pattern more closely in formal contexts, and by starting the sentence with an inverted "should you" construction, you are using a formal construction, which makes it more likely that we would follow this any/some distinction. In informal English, you could perfectly well say "If you need some further information...", though I prefer "If you need any further information". You could also say "Should you need further information..." or "Should you need any further information..."

Hello Michela, as someone said it's a fixed thing there. But, to put some perspective to it, any can mean some. Any doesn't have a distinction of an amount. Any can be one or some or all or none or half.. So, should you need any information, means if it is one more thing, ten more, or 536 more things, you'll help to provide that information. You're so willing to help, it can be however much that person wants or needs. Any/anything/anywhere/anyway/ etc, all lack a limit or care toward one choice or the other..

Because it is a question.

Or you could also say if you need more info please contact.. And then you can put down who you want people to contact

You mean do you need anymore information?