I don't think you need specialized software. If your questionnaire responses are quantitative, then SPSS, SAS should work just fine. If the responses are open-ended, then any of the qualitative data analysis packages such as Atlas-ti, NVivo, Max QDA etc. will serve your purposes.
Not sure if you are asking for tool for analysis of competed Delphi Survey (all rounds) or to facilitate each round of voting and process of arriving at final rankings. Are you looking to understand how voting at each round effected subsequent rounds of voting? Knowing the research question or if you are looking for tool to support the Delphi Survey process would help me answer the question.
Thanks for clarification. I've seen public/stakeholder participation platforms that automate the process and report the final result but nothing I know of is "branded" as giving Delphi Survey analysis (which is ranking of votes given to each option). I use an online survey service and most of these type services (e.g. FluidSurvey, SurveyGizmo) have ability to create the Delphi Survey and report final rankings.