Is it permissible for a person who has no access to water and finds it difficult to obtain it, and feels embarrassed by his family when buying water for ghusl, to perform tayammum?

It is not permissible to perform tayammum in lieu of ghusl for someone who can obtain water and does not fear harm from using it. A person's shyness towards their parents is not an excuse that permits tayammum, because Allah is more deserving of being shy before Him. And shyness is not among the excuses that permit tayammum in Islamic law.