Without more information, there are a lot of variables here:
For a
revenue flight:
If booked with bmi, on a bmi itinerary (or, at least, an itinerary
starting with a bmi flight), then
stewardo is quite correct, you will get a receipt as part of check-in, or can request one at the ticket desk - this is perfectly standard, and pretty much all e-Ticket airlines do it this way. If you provided an E-Mail address at the time of booking, I'd also have expected you to get an E-Mail receipt, and for a web booking you would have got both an on-screen, printable receipt, and a copy via E-Mail.
If booked with a TA, then, you are quite right, bmi refuse to issue receipts for TA bookings (and that includes TA's, corporate travel agencies, third-party web booking systems etc.). They are somewhat more inflexible on this than other airlines, e.g. both BA and LH will generate a ticket receipt on check-in for
published fares, generally as an option at OLCI or SSCI, but bmi won't and will refer you to your TA in this case (has caused me some problems with expenses claims in the past).
For an
award flight:
If the first segment is on bmi, you
may get a receipt at check-in, YMMV - you definitely should if it is an entirely bmi itinerary, but then, in that case, you should have been able to book online, and would have got a receipt from the web site.
Otherwise, you are correct, no receipt is generated (unless, of course, it isn't an e-Ticket route, in which case you will still get a paper ticket, and an itinerary/receipt, like in the good old days), which, I agree, is a king sized pain. There is, allegedly, an E-Mail address that the ICC can give you that will allow you to request a receipt - no, I've never got it to work.
It's an interesting question as to whether this breaches merchant agreements - I'm really not sure (though, thinking about it, if you order flowers via the 'phone and pay by CCD, you don't ordinarily get a receipt sent to you, and the flowers go somewhere else in any case....), what I do know is that most UK law on issuing receipts/invoices is centered around VAT, and air tickets are outside the scope of VAT, and thus not covered. If anyone is aware of any legislation that requires a company to issue a receipt for an otherwise non-VATable transaction in the UK, I'd like to know about it, as it would help me win a few arguments over expenses claims!
Willard the Bear - The only receipt I'm interested in is the BP that gets me onto the plane, however, when filling in expenses claims, Ken may have a different opinion.