Unless your professor assigns a specific topic or asks you to answer one question, you'll need to choose one. The Purdue University's Online Writing Lab suggests starting by brainstorming a list of possible questions. Put pen to paper -- or your hands to your keyboard -- and make a list of ideas. Review your list, looking for a topic that jumps out as more interesting or a better fit for the assignment than others.
The word "inquiry" itself refers to requesting information or investigating information in response to a question. The first step of your paper-writing process -- after choosing a topic -- is to ask a specific question. This doesn't always mean a scientific inquiry or a hypothesis that you form on a natural phenomena. Although inquiry papers can cover scientific and social sciences topics, they can also bridge other areas. For example, you might be writing an inquiry paper for an art history course. Your question may be, "How did the political climate and events surrounding World War I influence the cubist artists?" Note that the question includes specifics, not generalizations.
Turn your question into a thesis statement. This is the beginning of your paper, and provides a road map for the reader to understand the topic, why he should care about it and what your position is. This may change as you conduct your research, depending on what your sources say and what information you gather. Collect information that supports your thesis and answers your question. Use authoritative sources such as academic journal articles, your textbooks or websites authored by reputable experts and professionals. For example, your paper might focus on how maternal depression affects a young child's transition into school. You may choose sources such as the American Psychological Association or a peer-reviewed journal such as "Child Development." Integrate the resources -- and the information that they provide -- into an outline that details the order and arrangement of your inquiry paper.
Take your outline and expand upon each section, exploring your views on the topic and integrating the authoritative sources. Include in-text citations for your source information. This doesn't just mean adding an article name and author after a direct quote. You will also provide this information to the reader after citing a thought, idea or statistic that is someone else's. Provide a conclusion, results or discussion section to wrap up the paper. End the assignment with a bibliography that lists every source that you've used. The precise style that you'll employ depends on whom you're writing the paper for, what reason and what the requirements are. For example, you're likely to use APA style if you're writing a social sciences paper, MLA for a literature or arts focus and Chicago for either social sciences, history or the arts -- your teacher or publisher will likely stipulate.