scunner, By "very self conscious", do you mean she's talked about being uncomfortable with the hair or are you reading it in her body language? With the latter, I'd advise checking that is what she's self conscious about.
At that age, I didn't feel bad about my body hair (except when my mother would call me cactus legs or otherwise mock me for it), but I felt really awkward in swimsuits as I didn't seem to have the smooth silhouette others seemed to have/shown in media (I was underweight and used to feel like I had little boob bumps, rib bumps, and then sticky out hips), and it took some time to find that the right swimsuit for me is a one piece with short-like bottom, even now that I'm a healthy weight as I prefer not having any focus on my hips. I know others who were bigger than me who similarly found bathing suits seemed to highlight the parts they were least comfortable with.
For body hair, it was covered in some of the books my kids have and I waited until my kids asked. As neither their father or I shave, we didn't have anything in the house so I bought razor, extra heads, and foam. We talked through it & I shaved her the first time she sat in shorts in the bathroom. She only wanted her lower legs done at that point, but now she has her kit which she can use as she likes. We also watched this video on youtube, I think it's something like 21 ways to remove hair and had a laugh over some them - duct tape came up.
I try to make it as easy going as possible - I didn't want them to feel because their father and I do things our way that they shouldn't give different ways a try and see how it suits them -- just as I bought clippers when they wanted to maintain short hair even though their father and I both have waist-length+ hair. I don't get it, and I wouldn't be comfortable with my 13-year-old having those depilatory creams, but it's a pretty common thing to try out.