I have two rabbits and they have mites. I have to take them to vet, but I am not currently capable of buying a carrier. What can I do?
Guess this is an answer, though probably not the one you were looking for...
Find someone you can borrow one from.
Or get a good solid box and figure out how to attach that to the bike.
Or find someone with a car who can give you a lift rather than using the bike, which may be a lot less terrifying to the critters.
(I did transport a cat in her carrier on the back of my non-motor bike some decades ago, but it was a solid carrier, strapped VERY well to the bike, I moved as smoothly and cautiously as I could, and she was one of the most trusting cats I've ever known -- if I said it was OK, she was willing to take my word for it.)