is there a way to send email without any provider but by our internal SMTP
If your mail server can handle SMTP relaying and permits Reply-To then yes. There are many plugins that will handle that straight forwarding for you.
But that's really a conversation with your IT team and to be frank, some IT teams working with mail are er, not up to the challenge. ;)
Show them this part.
If your IT team has setup a local SMTP server, use that server. They probably also gave you a username and password and e-mail address to go alone with it. That's all you need.
If they can help you then you will be set. If they don't know how to do that (SMTP isn't that hard) then perhaps they can escalate to their mail software vendor for support. The rest of us here can't really assist you with that.