Is there a budget air flies between Yantai,Shandong and Singapore?

Probably, it depends on the airlines and the air fare change from time to time.
As for your situation, I'm afraid there is no direct flight for the journey. Thus you can take a flight to Beijing or Guangzhou, where you can catch a direct flight to your destination.
About the budget air, I suggest you refer to the following website to know better:
https://www.travelchinaguide.com/china-flights/