No, ordinarily an award letter would issue first. You shouldn’t have to pay a deposit before you receive an award letter - that’s just stupid. If you’ve been talking to admissions, I’d suggest talking to someone in financial aid, or vice versa. And if you’ve already done that, then go up the food chain to someone with more authority.
It makes no sense, though that your admission letter got lost . . . and I’m guessing that perhaps the same thing happened with the award letter. So they’re assuming you already have that letter even though you don’t. Have you moved or something that would have caused these problems?