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

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."

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)

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