“Ban” (番, or ばん) is a sort of generic classifier for ordinality. Sometimes you see “me” added (ichibanme, or 一番目), but I think it’s sort of redundant. “Shichi” does mean seven, but that is the on-yomi, or Chinese reading, and I don’t think I’ve ever heard it used in an ordinal. I’m not sure whether or not it’s actually wrong to use it, and people would probably understand what you were trying to say, but it would be more natural to use the kun-yomi (Japanese reading) for that particular number. So “seventh” = “nanaban” (七番 or ななばん, or alternately 七番目 or ななばんめ if you want to be completely specific).
As with many things Japanese, though, context matters. If you were referring to the seventh day of the month, for instance, you’d use nanoka (七日, or なのか) instead of nanaban.