Looks pretty straight forward, unless I'm not understanding something. You'd first need to change your method signature to take a zip code:
Function GetTimeZone(ByVal zipcode As Integer) As Integer
Then, you'd have to change your query to search by that zip code:
queryString = "SELECT Top 1 Zips.TimeZone "
queryString += "FROM Zips "
queryString += "WHERE (((Zips.Zip)=" & zipcode & "));"
Everything else should work as is.