Will I introduce survey bias if I do the following?

[Tenderfoot]

email was sent to respondents and now a thought has been posed to call everyone who either hasn't opened the survey, or have yet to submit their answers, in the hopes of getting more responses. I suspect that this will in fact introduce bias? True or not? If not, why not?
This survey is intended for current customers to fill out. This is not a random survey, will that affect the answer to my question? My gut is telling me that if a customer chooses not to do a survey, that is their "right". By calling them and asking them to fill it out, I'm not so sure that the data collected will be honest and have any true value?