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?)
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.
3 and 5
What injection(s) does a person need to receive by age 16?
All five variations of the questions are grammatical and convey the same meaning. You can use any of them to ask about the injection that a person who is 16 years old needs to receive.
Some other expressions you can use to ask the same question are:
1. What immunization does a 16-year-old need?
2. Which vaccine is recommended for a person who is 16 years old?
3. At the age of 16, what injection should a person get?
These alternative expressions can be used interchangeably, depending on personal preference or context.