Survey Monkey is a great tool we use for sending out surveys. It can be integrated with a MailChimp account to send the survey out to.
https://help.surveymonkey.com/articles/en_US/kb/Editing-Questions?bc=Creating_Surveys
Most of the time you will only need to use the free version of Survey Monkey. Ensure you have set up an account for your client (don't use ours account to send out a survey for a client). The free account allows you to send unlimited surveys with 10 questions and gather up to 100 responses. You can't customise the colours and the logo but most of the time that is fine, especially if you are sending it out via MailChimp.
To view paid plans, go here: https://www.surveymonkey.com/pricing/?ut_source=megamenu