There is a setting that can be turned on in Notes in the User Security dialog. (You didn't mention the version of Lotus Notes. The dialog may be different if you're on an older version.) The setting is:
Don't prompt for a password from other Lotus Notes-based programs (reduces security)
.
Those last two words may be in parenthesis, but don't let that fool you. They are pretty important! The setting allows all programs running on that user's computer to use the Notes APIs without authentication. That opens the door for an email-borne piece of malware, amongst other threats. And never mind the fact that such threats have been rare in the Lotus Notes world. A big part of the reason for that is that keeping this setting disabled is the default.
From what I've said above, I'm sure you can understand why there isn't an easy way to do what you want to do!
The only way to disable the password prompt specifically for your application will be to use the Lotus Notes C API's feature set known as Extension Manager. I.e., you write a C source file that is compiled into a DLL. Your DLL must be installed on all users' machines, and an entry must be made in the notes.ini file in order to have it loaded when the Notes core DLLs run. The help files for the C API include some sample code showing how to do the Notes-specific part of this. You'd have to figure out the details of identifying the fact that it was invoked for your application, and (if I remember correctly) you would also have to write code to prompt the user for their password once and be responsible for securely storing it somewhere where your DLL can retrieve and decrypt it whenever it needs it.