Stanley Metcalf was found fatally injured on Church Lane in Sproatley, East Yorkshire, at around 4pm on July 26 while visiting relatives, Humberside Police said.
Stanley's funeral is due to take place tomorrow. (Weds August 22)
His family said in a tribute statement: "“We are absolutely devastated and completely heartbroken without Stanley and it still doesn’t feel real we won’t see his happy, smiling face again.
“There are no real words to express how the loss of Stanley has affected us and we all miss him so much.
“He was so vibrant and full of energy, a typical six-year-old boy who just wanted to have fun and enjoy life, and he adored playing football.
“The condolences and support we have received has been incredibly touching and we would like to thank everyone, friends, the general public and the media, for their kind words and sensitivity during what is such an emotional and devastating time for us.
“Stanley’s funeral will be a day dedicated to him and for him, he is all that matters.
“We would please ask that our wishes of privacy are respected and his day remains about him, with those attending who knew Stanley being allowed to mourn alongside us without any intrusion.
“Stanley was a loving, caring and beautiful boy, and there is huge gap in all our lives now that will just never be filled.”
Police were called to a property on Church Lane in Sproatley at around 4pm on Thursday, 26 July following reports Stanley had been seriously injured at the property.
Stanley was immediately taken to hospital where he died the same day.
A Humberside Police spokesman said: "An investigation is ongoing and various, voluntarily interviews have been conducted by officers to assist with enquiries. No arrests have been made."
An inquest into Stanley's death was opened and adjourned at Hull Coroner's Court earlier this month,
Senior coroner Professor Paul Marks was told that a post-mortem examination carried out on July 30 had revealed the boy’s cause of death to be “airgun projectile wound to abdomen”.
Giving evidence, Detective Sergeant Julian Gibbs, of Humberside Police, said that the force were called to the scene of the incident after receiving a call from the ambulance service.
He said: “Upon examination by family members, Stanley Metcalf was found to have a small hole to the side of the abdomen.”
Shortly after his death, the force said early indications pointed to it being a “tragic accident involving a pellet gun”.