It would be best not to query an agent until you have three picture books perfected as well as you can. You'll only query one manuscript, but any interested agent will ask to see more, and you'll need to have them ready.
Don't query directly to publishers unless you are sure you won't want an agent for that manuscript. Because if you shop the manuscript yourself, an agent won't want to represent it. In general, if you envision a career as an agented author, you want to query agents, not editors. If you can't find an agent, you can always shop those manuscripts to houses that don't require one (and then query agents again with later work). But it doesn't work the other way around, because if you collect publisher rejections on a book, you've removed the agent's marketing options for that book.
Read the contest rules carefully. Some will state whether or not entries can currently be under consideration by agents or editors. If you fulfill all the contest rules, I'd suggest entering the contest as long as it's legit. While you wait for results, you can be working on more projects with the goal of eventually querying agents.