"Illegal" is quite a strong word but yes, the vehicle should be as described. Unless as part of the haggling something else was agreed, the vehicle should be what was put in the advert. If the advert says that it has 1 year MOT/Tax, then that is what you should now have (or at least 11½ months worth). It would have been easier to have checked before driving away though.
If you contact the seller with an agressive tone, using words like "illegal", the seller is unlikely to cooperate much. Try phoning the seller and politely pointing out what has happened. It is possible that the seller bought the tax disc but forgot to put it in the window. Whatever the reason, a decent seller will either send you the tax disc or a cheque.
If the seller refuses and if you have details of the MOT in writing (like a copy of the advert), then you would stand a strong chance of success in a small claims court. However, it is always going to be much better for both of you to sort the issue out between yourselves.