As ToadMum has indicated, these problems are more common with a family's first primary school application!
I know that some LAs will treat a late application as timely if there are very exceptional circumstances, e.g. single parent hospitalised at the crucial time, and no one able to act in loco parentis. An exception might also be made if it could be shown that the LA had done something incorrectly (e.g. if it lost a paper application that had been submitted by recorded delivery, and signed for, prior to the deadline).
Unfortunately, being unaware of the deadline may not count for much when the information would have been available on the LA website.
There is a responsibility on the LA to publish key information - and, I'm afraid, some responsibility on parents to be proactive.
If they contact the council and ask for a paper form, will they be treated as a late applicant? Does this mean that they would only be considered after all the people who applied on time.
If they applied now, I assume their application would be considered in the second round of allocations.
Probably a good idea for them to get their application in as soon as possible now (the longer they leave it, the worse it might look to an appeal panel), and to attach a very apologetic note explaining the circumstances (in the hope that these might be viewed with a bit of sympathy - if not by the LA, then perhaps by an appeal panel).
I see no reason why they can't appeal for a grammar school - in due course. They would be appealing against non-qualification and probably (depending on local circumstances) oversubscription as well.