Simple answer : Yes
I would assume that your contact form mails to a limited list of people, and doesn't take a 'to' e-mail address from the form itself. (you might take an 'about' (subject) heading from the form, and translate that behind the scenes to the appropriate e-mail address, but don't accept a tainted e-mail)
If so, the only people they can spam are your company -- and when you have someone flooding one mailbox, it's annoying, but easily deleted. They'd have been able to have done it just as easily by getting an e-mail address of an employee there, or waiting for you to respond to the form submission, and then flooding you.
I don't know your exact situation, and why you believe there to be this need, but I don't think CAPTCHAs are the way to go (especially because they needlessly exclude people with poor vision). I'd suggest instead taking a look at Limit submissions over time?.
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