You may be confusing Stamp Duty with the Cost of supplying a Contract Note.
A Contract Note is the legal record of the transfer of ownership of shares. Originally issued in paper form, they are now most often issued in (digitally-signed) electronic form.
Governments being governments, decided to tax share-dealing (among other activities), and came up with the idea of Stamp Duty (thought to have originated in Spain before being (re)introduced across much of Europe during the 17th century).
The name Stamp Duty originated when it was common practice to afix a Revenue Stamp to official documents and contracts to record that the appropriate tax had been paid. As the second link states:
Governments enforce the payment of the tax by making unstamped documents unenforcable in court.
Therefore, the terms of a Contract Note could not be enforced without having paid the "stamp duty" and having a revenue stamp attached to it.
Nowadays, of course, most contract notes are delivered electronically, and would usually be digitally signed. Although these will not have revenue stamps (after which the tax was named), the tax must still be paid.
Therefore, when applicable1, all share-dealing will involve paying the associated Stamp Duty.
The page on Contract Notes includes:
How to Get it?
[...] Now a day’s almost all broking houses provide digital contract notes. This service is normally free but at times your broker might charge a nominal fee.
So, in some cases2, you may be charged a (hopefully small) fee to get a copy of a contract note, but such fees are unrelated to Stamp Duty.
You mention in a comment NSDL and the possibility of "No stamp duty on transfer of securities;". This company sounds like a Nominee Firm where shares are "officially" held in the name of the nominee (e.g. NSDL) and they "internally" record that individual investors are the "true" owners.
If this is the case, then it may be possible to avoid stamp duty, but – I suspect – only if the transaction is between two people who are both members of NSDL (because in this case, the shares aren't "really" changing ownership as far as the "outside world" is concerned)3.
1 Exactly when, and under what conditions, Stamp Duty must be paid will vary with jurisdiction. For example, some places may waive it for small transactions.
2 Since it is, in essence, a legal document, it's possible that some jurisdictions have regulations that make charging to issue a Contract Note illegal.
3 Presumably, since the NSDL website makes that claim, it is possible to avoid Stamp Duty in India if there is no "external" change in ownership. However, I suspect that some jurisdictions may not allow you to avoid Stamp Duty in this way. As always, check with your local laws and regulations.