My 15 year old son uses Gillette Fusion. So does my husband. I've used it to shave my legs and under my arms and it's great. Of course, I know how to shave my legs, which might be the difference in hurting myself or not.
What I use most though is an man's electric razor, the kind with a pop-up sideburn shaver. The pop-up pushes straight up in the same direction as the shaver (not out perpendcular to the shaver.) The reason I mention that aspect is that this is the part of the electric shaver I use most to shave my legs. I really like using an electric shaver. I don't get the itchies when the hair is growing back in, or red bumpy rashes, and I don't cut myself either. I can shave everyday with an electric razor if I want to; I can't with a regular shaver because of the itchy bumps, no matter what razor I use. I guess I've just got real sensitive leg skin. I also learned a long time ago that women's razors don't work for me. They aren't strong enough, LOL's!
Of course, her first time she might need to shave with a real razor because of the amount of or length of leg hair, but if she's really squeamish and you're really worried, using an electric one might be the right answer for her after the first time. If you go that route, try to get one that you can charge up. It's so much better than having to sit beside an electrical outlet on the floor because it has to be plugged in!
At some point in your girls' lives, they aren't going to be able to do everything together. When your 16 year old wants to date, you aren't going to let the 14 year go on dates too. It's better to separate puberty and growing up issues now, before more issues come up. She'll have to learn it soon anyway, because her older sister will soon have her period, and she can't have that experience early. Maybe this is the time to talk about the facts of life with her, so that she understands a little of what her sister is going through, and that's why she has to wait for certain things like shaving her legs. If she really wants to share her sister's leg shaving thing though, and does it without your approval, maybe it's not the end of the world, and after she does it once or twice, maybe she will forget about it until she's older.
Good luck and bless you for all that child rearing you are doing, whew!!!
All my best, D.