All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
You should not allow that teacher to come in your house unless there is another old person other than a child since staying alone with a non-Mahram man is not permissible. The existence of a child does not change this ruling. Imam An-Nawawi said: “It is forbidden for a man to be in seclusion with a non-Mahram woman according to the agreement of Muslim scholars. The same ruling applies if there is one from whom they do not feel shame.”
If by “to accompany him for a school vacation” you mean to travel with him, then it is strictly forbidden since a woman is not permitted to travel alone without being accompanied by a Mahram who can protect her and take care of her.
The reason for this prohibition is that it leads to many evils such as being alone with a non-Mahram man, looking at him, mixing with him etc.; all these matters are forbidden in the Sharee'ah. There are many Ahadeeth prohibiting a woman from traveling without being accompanied by a Mahram. Ibn ‘Umar narrated that the Prophet said: “No woman should travel a distance of three days unless her Mahram is with her.” [Al-Bukhari and Muslim]
Abu Hurayrah narrated that the Prophet said: “It is not permissible for a woman to travel a day and night journey except if a Mahram is with her.”
Many scholars stipulate such a condition in the journey for Hajj and ‘Umrah; so such a ruling, with greater reason, should be applied to a picnic trip.
For more benefit on the prohibition for a woman to travel without a Mahram, please refer to Fataawa 262621, 88562, 193077, 102095, and 85945.
Allah knows best.