The answer is simple: Get SMTP support. Never have a site without it.
Either use hosting smtp server or 3rd party service. And setup SPF and DKIM.
1. There are maybe 1000 different websites on the same web server (probably more). All of these use the same IP address. If any of these websites send out spam (ie have been hacked and are now misused in this way) the IP address will end up on various lists of known spammers. Many incoming email servers discard letters from blacklisted IP addresses.
2. Some incoming email servers check which IP address a domain uses for incoming email and give minus points if the web server’s address is far from that address. (Here SPF can help, there is a declaration in your DNS file about which IP addresses can send e-mail on behalf of your domain)
3. Some web hosts block the distribution of e-mail from the server.