Does AESA need to send me any documents or acknowledgement of receipt when notifying the Declaration of Continued Airworthiness to renew the NARC

No, AESA is under no obligation to respond or send any acknowledgement of receipt of your statement. 

However, the owner or person responsible for the maintenance and maintenance of airworthiness must keep the proof of the notification as proof that the notification has been made. 
 
If due to the impossibility of carrying out the online procedure, the declaration has been submitted by 

SOLICITUD GENERAL, you will receive an email to the address indicated therein once your return has been registered.