Actually, you probably have more experience that you may realize. You mentioned being a camp couselor, and I'm sure, in many ways, your duties there in many instances were very much like the same duties of a tutor. I would address this directly in a cover letter attached to your resume, or in an email sent with your resume, by stating something like, "although I don't have traditional or formal tutoring experience, my extensive experience as a camp couselor entailed and encompassed many of the same skills required by a tutor."
Then you could go on listing specific examples of how you've "counseled" or "tutored" kids. For example, if you think about it, what's tutoring really? It's sitting down with a child, and being able to deliver a lesson to them. How do you do that? You need to be keen and be able to recognize their strengths and weaknesses, and tailor your lesson delivery method to that. I mean, isn't that sort of the same thing as you did as a counselor? Maybe the "lessons" at camp weren't math or reading or science or such, but they were indeed lessons. Whether they were about time management (being on time for certain activities), or conflict resolution (working through problems with roomies or friends or even enemies). Or even helping with social and interpersonal skills, whether in/with groups as a team, or even individually as a self esteem builder.
I'm sure you've done many of those things, all you have to do is get creative and figure out which/what you'd like to highlight, and show them how it relates to "tutoring". That way, not only do you show that you do indeed have experience, but you also show that you understand that tutoring, is NOT just about sitting down and teaching someone how to add two numbers or such. You know it's about much more. You know it's about helping someone secure confidence in themselves, you know it's about teaching them to connect the dots so to speak, so that they can figure things out for themselves, when you're not there to guide/teach them. And so on and so forth.
So technically, you couldn't lie even if you wanted to, because you've already had the experience. ...Good luck!