Don't use brushwood killer, you won't be able to plant anything at all for at least six months, probably a year. Digging is the best option - cut the grass down and if its solid, like turf, cut and remove it, if its clumps, dig them out and turn the whole thing over.

You might want to find a friend who hasn't got a bad back to share the allotment with as growing veg. requires a lot of digging.

'It depends on what kind of grass is. If it is couch grass and you are not prepared to use chemicals, you better give it back and wait for a better block to become available. By digging or rotovating you will only make things worse and more will grow. If it is only garden grass cover it with an old carpet or black plastic sheet and will all die very soon. I had an allotment many years ago and took me five years to clear the couch grass by digging and hand collecting the roots. You leave a few roots behind and it grows back to cover the whole area.

Can you borrow a strimmer to cut it back to a managable length?