Why does my contact form not work? I am not getting emails.

I was having the same problem but fixed it by following Meg’s older post that says:

The Email Will Not Send From Another Domain
Are you finding that all is well when you send from john@example.com to the recipient address fred@example.com but not from jill@anotherdomain.com? It is likely that your email provider is not letting you send from any email address. When the contact form sends the email it will use the email address of the form filler as the ‘From Address’ in the email. Some email providers will not allow this because there is a possibility for abuse by spammers. If this is the case then you will need to fix the from address to one within your domain. On the settings screen of the contact form tick ‘Override ‘From’ Address :‘ and then enter an email address that you own in the ‘‘From’ Email Address :‘ box underneath. This will make all emails appear to come from you instead. You won’t be able to click the ‘reply’ button to quickly reply, but don’t worry, your form-filler’s email address will be in the message so you will know who to reply to. If you are not hap
py with this and want to be able to click the ‘reply’ button, consider using a third party SMTP provider along with an SMTP plugin like WP Mail SMTP for example.

That was an old post. It does support reply-to header.