I have been involved in development of both client/server and web applications over the past several years. Working as a team lead helped the development team and business users in understanding the requirements/translating.
Based on my understanding with the brief description, all you need is a web application where you can maintain questions / set a questionnaire what I call as administration. On the other side you need participants to take the exam based on the questionnaire you set.
Note: With the limited budget we may not be able to completely automate the administrative tasks and may have to do things manually.