Is it permissible to sacrifice a camel, from the son's private funds, as an act of charity on behalf of his deceased father? And is it permissible to substitute that with the value of the charity to be spent on building a mosque or other ongoing charity?

If you pledged to your father that you would fulfill his will, then you must honor this pledge to the best of your ability, as Allah the Almighty says: "And fulfill the covenant; indeed, the covenant will be asked about." This fulfillment is considered part of perfecting filial piety towards parents after their death.

However, if it was merely a will you did not pledge to fulfill, then its execution is a voluntary act for which you will be rewarded. If you wish to execute the will from your own money, you can convert it into an ongoing charity (Sadaqah Jariyah) such as building a mosque, a school, or a hospital, or providing water to those in need, or distributing the value to the poor and needy.

Ongoing charity is better than intermittent charity; as the Prophet, peace be upon him, said: "When a person dies, his deeds come to an end except for three: ongoing charity, beneficial knowledge, or a righteous child who prays for him," and his saying: "Seven things continue to earn reward for a servant after his death while he is in his grave: one who teaches knowledge, or digs a river, or digs a well, or plants a palm tree, or builds a mosque, or inherits a Qur'an, or leaves a righteous child who prays for him after his death." Therefore, ongoing charity on behalf of the father is recommended.