Yep, I had exactly the same issue and it took me a fair amount of time to work out a way around this common issue.
AWS Instances and Lightsail, don’t allow the usual PHP way of sending forms, so you need a work around.
I eventually found WP Mail SMTP to solve this issue for me, it uses SMTP to send a mail via an email account. It works with most providers and has special Google Mail, Mailgun and Sendgrid. I used a spare SMTP account from our hosting provider to give us this functionality.
Just follow the simple instructions from top to bottom, make sure the settings are correct for the SMTP settings. Once complete, scroll back to the top and send a tester email.
I did run into a few issues at first and had to drop my password into the wpconfig file using:
define( 'WPMS_ON', true ); define( 'WPMS_SMTP_PASS', 'your_password' );
Once working all of your contact form 7 submissions are sent to the email address specified on the form.