Kurt,
The issue you're running into is authentication, or basically "How do
other mail servers know that my mail server is allowed to send
email?". This comes up because most mail servers will reject messages
they think are from spammers.
It's generally pretty common to store auth credentials (user/password)
on a server, but I understand your hesitancy. I wouldn't use your
personal account. I would setup a separate account for the website
you're creating.
Also, you only need an application specific password if you're using
2-factor auth for gmail (which you should setup anyway).
If you don't want to put your username and password on the server, you
can setup reverse DNS, SPF and DKIM for the domain name that's sending
the email. Then when you send email, the receiving server can check
your DNS records to make sure that your ip address is allowed to send
email. Here's a good article on it:
https://wiki.zimbra.com/wiki/Best_Practices_on_Email_Protection:_SPF,_DKIM_and_DMARC
Let me know if you have any other questions, and good luck!