I would use "vaccination" instead of "injection."
I would also rephrase: What vaccinations does a 16 year old need to receive?
All your sentences are grammatical, but #3 is awkward.
1. What injection does a person at 16 years need to receive?
2. What injection does a person at 16 need to receive?
3. What injection does a person at 16 years old need to receive?
4. What injection does a person aged 16 need to receive?
5. What injection does a person at age 16 need to receive?
(Which ones are grammatical? What other expr4essions can we use?)
2 answers
3 and 5
What injection(s) does a person need to receive by age 16?
What injection(s) does a person need to receive by age 16?