Most derailer sprockets are made of plastic. They won't hold up to the stress of an engine drive chain. The photo you show is a steel industrial sprocket available at industrial suppliers. It will take some fabrication skills to adapt one to your needs.
Derailer chain is slightly narrower than coaster type.
Idler sprockets are available from McMaster Carr.
I have welded sprockets to modified V belt idlers.
One suggestion: the strongest mount would bridge two sections of the rear frame instead of just sitting on one rail.