Hi Jane,
To answer your original question, No this could not happen, and as to why is explained on page 4 of the document RedFraggle linked.
I realise the science is quite complicated so I'll try and make it easy, though feel free to ask questions, and I'm sorry if I've 'dumbed it down' too much.
The new virus is a GMO (genetically modifed organism) that is the myxoma virus which has been altered to carry the important bit of the RHD virus. The myxoma virus has had the genes removed that allow it to replicate in the rabbit. In effect, the scientists have taken the myxoma virus, removed the bits that allow it to multiply, and stuck in a bit of the RHD virus that then allows the rabbits immune system to see both the myxoma and the RHD viruses, make antibodies, and develop immunity. The dosage given at the top of the page is the dose that will allow the rabbit to develop immunity but not get sick.
If a bug bit a recently innoculated rabbit, then bit a non-innoculated rabbit, it could pass on some of the myxoma-RHD virus, but as this virus has no way of replicating, it would indeed act like a mini-vaccine. Though would not be enough for the rabbit to develop full immunity.