Yes, you can do this, you can open a 3-in-1 account (which is Bank Account + Trading Account + Demat Account) with ICICI Bank (ICICI Direct), HDFC Bank (HDFC Securities), SBI Bank (SBI Securities), Axis Bank (Axis Direct) and so on.
With Banks listed above, you can get direct access to apply in IPO on a single click. You don't have to submit an application via ASBA. These banks are part of Self Certified Syndicate Banks. They are providing you a platform where in single click you can apply in an IPO.
Advantage: This process is quick and easy. You don't have to log in to your bank portal for ASBA.
Disadvantage: Charges are very high compared to other brokers. Usually, retail investor applies or get one application which will be around 14 to 20K price range, selling an IPO script with such banks/brokers will take your good amount of profit. ICICI Direct, HDFC Securities, Axis Direct, SBI Securities charge approx 0.5% brokerage on selling of these scripts in equity delivery trading.