Welcome aboard, sess4561.
A word for your boss - he had better do something to make sure the user is
aware their information is being shared with a third party. Even if it's not private sensitive info - which it is, you're passing off the email and phone - this is a good way to kill user trust and/or get sued. What they are doing here is taking away the user's right to choose where their data does - or does not - go.
With that in mind, we can take it out of the closet and use it as an upsell: "Get even more buyers for your home, submit to X properties." Everyone wins.
That being said . . . if the white path is chosen, the solution is obvious. :-) If not, an alternative to adding another layer of technology (which will fail if JS is disabled) requires one answer:
.... waits 10 mins or so to see if the customer clicks 'yes' 'no' 'maybe', and then submits it to the other companies database?
The question is, does the user's response affect the disposition of the submitted data? That is, if they say "no" the third party gets no info? If this is true, modify the below accordingly.
I'd do this via a simple cron, completely transparent, and you'd only have to run it once an hour. You would add a field to your database, boolean or tinyint, and once sent, it gets marked to 1 or true, so it doesn't get sent again.
To make it **really easy** you'd just ask them for the email to send the data to, and compose a mailer as the cron. Open the DB, get leads where sent=0, done, run it once a day, once an hour, whatever.
If they want the thing submitted through the form, just curl it with the data as presented.