I want to start a new SIP from ICICI Bank without disturbing the SIP from HDFC Bank

Yes you can start a SIP from your ICICI Bank account, without disturbing your existing SIP which is being debited from your HDFC bank account. Firstly, you have to identify the mutual fund scheme where you will start your SIP. Next in the SIP ECS form of the selected scheme, mention your ICICI Bank account details. Your new SIP will be debited from ICICI Bank account. If your KYC is updated, then you do not need to submit fresh documentation for your KYC. If you are doing your SIP online, check if the KYC is updated or alternatively, you can check with your financial advisor.