If you want the email to be sent from the account of the user of the app, a proper authentication would be required.
As a result, either you could send an intent to email app (as you mentioned) or you could do it yourself (as in second method mentioned by you).
But the second method obviously requires password since you are sending an email on behalf of the user (from their account). Gmail (or any mail service provider as a matter of fact) won't allow that without a password.
So to answer your question in a nutshell, no. You can not do it without a password.
In stead, you could use a work around. You can send all emails sent through your apps from a single mail id (which is created by you so you know the password). Now in the content of the email, you can store the email id of the user from whom you are asking for the feedback.
This way, you just have to ask for the mail id (no password) and you also get their contact information if you want to contact them back.
Hope this helps.
Good luck.