Vashishtha is considered as the family priest of the Raghu dynasty and teacher of Lord Rama and his brothers.
This is what Wikipedia says about him
In the Ramayana, he was the family priest of the Raghu dynasty and teacher of Lord Rama and his brothers.
It seems like Vashishtha most notables work is in knowledge-work - for example, Vashistha is credited as the chief author of Mandala 7 of the Rigveda.
Since he is considered as the family priest too - his role mostly seems to be around teaching Vedic values and other subjects more related to academic education of literature and not much about war skills such as archery and other weapons.
Not sure if it was Vashishtha who taught Rama and his brothers the weapon skills.
Next, Vishvamitra - he too is mentioned as the preceptor of Rama and his brother Lakshmana. While there is a mention that Vishvamitra is the one who gives them the knowledge of the Devastras or celestial weaponry [bala and ati bala], - he came in Rama's life much later - and by that time - Rama was already well versed with his archery and other weapons skills.
In Mahabharata, There is absolutely no doubt who taught archery to Arjuna.
Do we have any such reference which clearly mentions Who taught archery to Lord Rama?