I think the only way you can do it with the built-in contact form page is to create one field per radio button and not use a label for any of them. Trouble is, this will just produce a long list of options when what you probably want to do is have them in columns or laid in another, more elegant way.
Will Woodgate has some really useful stuff on hacking the contact form here
But in the end you might be better off with a third party solution. Have a look at the Addons section in Rapidweaver Community and look for forms stacks or forms plugins.
There are also some external services that some people here swear by but those may be a bit overkill for your needs. A Stacks-based form builder will certainly be able to do what you want.