6month Baby apply for student pass

A student pass requires full-time course attendance.
I am not sure a 6-months-old qualifies.
Ask at the child care centre where you want to enroll him/her.

another question who can online submit for the student pass? Me or the child care center ?

You need to apply for your child.  Other details, can be found in below links:

https://www.ica.gov.sg/pass/studentpass/apply

cos my infant care said they are willing to help me to apply for the student pass but will incurred some admin charge, so I wonder who should be the right person to apply for the pass..

When they said to help you to apply, I think they will do admin works like what an agent does for a working pass but the ownership should be at parents side not a third party. Read out below link for step by step process to apply (by YOU):

https://www.ica.gov.sg/apply/studentpass/msf