If at all possible, be very careful about scaling the results of your analysis. Non-response tends to be related to interest. For example, regardless of demographic match, people who fill out a survey about bus service tend to be people who are more interested in taking the bus than the average person. Therefore, estimates based on the survey respondents themselves tend to be much too high.
For example, the old rule of thumb about predicting consumer product sales from a standard 5-point buying intent question was: 75% of those who "definitely would buy", 25% of those who "probably would buy", and 5% of the rest. [Advertising Research Foundation Arrowhead project]
To directly answer the question, you can certainly do the logistic regression, but it is validly applied to the population you are analyzing (i.e., the population of survey respondents). Anything beyond that is an inference, similar to inferring opinions in Minnesota from surveys in Iowa and Wisconsin. That inference can be supported by some of the strategies suggested above, but is still an inference.
In many cases you MUST do the inference. The client paid for the survey, and wants to get at least some useful information out of it, so you have to give it your best shot, with whatever caveats you deem necessary.