Check your server IP or domain hasn't been blacklisted - we recently had an issue with Spamhaus listing our new server IP address and it immediately stopped all emails from sending via the webforms. Turned out there was no reverse IP (PTR record) set up for the server but now that's been done we've had no further issues. You can check blacklisting here: https://mxtoolbox.com/blacklists.aspx
Also, are the emails sending from an email address using the same domain as the website the form is on? For example, if it's a contact form do you have it set to come from e.g. "noreply@yourwebsite.com" or does it come from "sender.name@theirdomain.com"? If the latter, this isn't allowed by most ISPs now because of spoofing issues. If your webform is on www.mydomain.com/contact.php then you must set the email script to send from anyaddress@mydomain.com.
