Electronic questionnaires (questionnaires that are administered over the internet) are one of the best tools for social science reserachers today. They have many strengths and only a very few weaknesses.
The first potential weakness is the difficulty of getting a random sample. Internet access (especially broadband) is still not a universal thing. There are certain demographic groups who are less likely to have internet access. Therefore, it may be difficult to get a good, random sampling using an electronic questionnaire (because people from those groups with lower internet access will be underrepresented in the sample).
Second, electronic questionnaires (like paper surveys) do not allow for in-depth questions and answers as well as personal interviews do. Electronic questionnaires are much better suited to questions that can be answered "yes" or "no" or on a scale of 1 to 5. They are not suited for research in which the researcher wants to be able to interact with the respondent and probe his or her attitudes more deeply.
Overall, though, these weaknesses are not generally a big problem. Social science researchers make extensive use of such questionnaires today.