pneumococcal 13-valent is very expansive. The govt 母嬰院 offer the SAME, but only cover for children up to two.
So I think your younger one can go there. You have to bring his VACINATION records , e.g Hepatitis A (1st shot at 12mth, and 2nd on 18 mths)you can work back if it's missing, and they are free of charge.
FOR influenza, both of them should be coverred for free shots. GO ASK AH.