Hi!
This is my first contribution in the forum, and I'm interested in this issue.
In fact, when you use 'his', you exclude some people from the attendance. I mean, with the possessive adjective 'his', you mean that all people in the hall are men. It's better to use their, lest you might be considered gender-biased.
I speak from a linguistic point of view. I'm a teacher of English, but it is not my native tongue. So there might be something regional related to your country and different from my approach.
Some liguists think that 'their' is less formal, in this way you can use 'his/her' instead.
Your sentence would be
When the speaker sat down, everybody in the hall stood up and clapped his/her hands.
I hope this is clear and useful enough.