Possibly not on really cheap stuff but I would have thought even a lower end Trek would have a replaceable dropout
Very few bikes I know of have replaceable dropouts. Maybe a few specialist frames that allow for belt drive, or allow you to swap between track ends and standard vertical dropouts.
I suspect you're thinking of the derailleur hanger, on the drive side only, which can often appear to form part of the dropout. This is also most likely what the LBS has broke.
Get them to pay for it! Especially if they've already admitted they accidentally broke it.