They may be sending test messages to mailboxes they monitor to try to determine if your site can be used as a spam gateway. That's my suggestion.
Another possibility (as mentioned already) is that they're trying to improve the IP address reputation by sending apparently-good mails from the server.
As far as trying to block them is concerned - their bots are normally very stupid. They don't parse the HTML properly and they certainly don't read the form or support CSS. Add a field called something like 'email' inside a display='none' element. Ensure that this field is left blank by legitimate respondents, and bots are almost certainly going to fill it in anyway. Rename your real email field to something else.
Then any form submit with a value in the unfillable field is definitely spam and can be ignored.
Other options include using Javascript (which bots generally don't execute) to fill a hidden field, but these place more requirements on legitimate user agents (still better than a captcha though).
Provided the technique you use isn't too common or similar to those used by lots of other sites, the bots will not have a work-around for it.