Is it possible to have meningitis or encephalitis with no fever or vomiting?

No, fever, headache,: neck ache/stiffness, lethargy, changes in sensorium, activity & appetite & often vomiting are signs of these infections. You'd be pretty sick if you had meningitis or encephalitis. Even mild viral meningitis would be associated with fever & feeling pretty rotten.

Yes: This is not the typical presentation, but meningitis could present with listlessness and altered mental state without fever and vomiting. The key here is ill appearance in a patient who does not improve their mental state with a dose of motrin or tylenol (acetaminophen).