There are around 5.3 billion Internet users, amounting to 65.7% of the global population, and web technology is the backbone of the services delivered via the Internet. To ensure web applications are free from security-related bugs, web developers test the server-side web applications before deploying them to production. The tests are commonly conducted through the interfaces (i.e., Web API) that the applications expose since they are the entry points to the application. Fuzzing is one of the most promising automated software testing techniques suitable for this task; however, the research on (server-side) web application fuzzing has been rather limited compared to binary fuzzing which is researched extensively. This study reviews the state-of-the-art fuzzing frameworks for testing web applications through web API, identifies open challenges, and gives potential future research. We collect papers from seven online repositories of peer-reviewed articles over the last ten years. Compared to other similar studies, our review focuses more deeply on revealing prior work strategies in generating valid HTTP requests, utilising feedback from the Web Under Tests (WUTs), and expanding input spaces. The findings of this survey indicate that several crucial challenges need to be solved, such as the ineffectiveness of web instrumentation and the complexity of handling microservice applications. Furthermore, some potential research directions are also provided, such as fuzzing for web client programming. Ultimately, this paper aims to give a good starting point for developing a better web fuzzing framework.