Is it possible to get a tonsil infection after having your tonsils out?

Yes. There are "tonsils" that live behind the tongue called "lingual tonsils". These can also become infected. It is unusual, but does happen.
No-unless... Can't get a tonsil infection if you do not have any tonsils. Only thing is if the surgery did not remove all of the tonsillar tissue.
No. It is very rare to get a primary tonsil infection after the removal and recovery from tonsil surgery.