As @TomCallahan mentioned, HTML forms are not supported in all email clients. And even in the ones that it is supported, you will usually get a warning or other pop up from the email client warning you that this form is pushing information to an outside web page. Or the form displayed is not styled or displayed as you set it. For instance, Hotmail will display what is essentially a 'plain text' version of the form.
Basically using an HTML form inside an email is a terrible User Experience in the majority of email clients - regardless if supported or not.
Support of HTML forms in email: (as of 2017 - most recent display of support I could find)
My recoomendation is to either have the survey live on a Cloudpage or other website (survey monkey, etc.) or to have 1 question Yes or No response and have a cloud page that collects the button pushed and inserts into a DE. E.g. have the answer parameter added on URL in href.
<a href="https://yourURL.com?a=Yes">Yes</a> Then just use RequestParameter() on the Cloudpage to collect the value and insert into the DE.