Yes, I think so. Let me summarize;
A visitor fills out a Ninja Form and on submit Ninja Form is creating an email FROM xx@gmail.com TO xx@gmail.com. Is that about right?
In that case, use an SMTP plugin (mine or any of these) to delivered this email via smtp.gmail.com.
My guess is Gmail is dumping this email into a black hole because it recognizes the sender, but not the sending server, and is rejecting the email in an attempt to prevent forgery.
I would see the submission inside the WordPress site but it was never sent to my email.
That's almost correct; it's actually being sent, but it gets dumped at the receiving end. An intentional 'fumble' in American football terms.
