is this sentence correct : the audience were so excited that they all stood up clapping ? （I'm not sure which word to use here, clapping or to clap）.

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It's fine either way, although "clapping" sounds a bit more natural. By the way, we call that a "standing ovation".

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I would write:

"...that they all stood up and began clapping."
"...that they all stood up and began to clap."

Either would work. I can't think of any obvious difference between them, and both sound natural/normal.

You could also write "...that they all stood up and applauded."

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The audience was so excited that they received a standing ovation.

The audience was so excited that everyone stood up and clapped.

The audience was so excited that they received a standing ovation. (They is the performer here)

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Hello. I would write The audience was so excited that they all stoop up to clap.