I don’t think you need to drop her from the bridal party at all.
From your questions, I’m assuming the bridesmaids’ dresses are sleeveless or strapless or in some style that exposes arms. If that’s the case, you can always let your sister wear a long-sleeved version of the dress (if it is available), or even just use a matching bolero that will cover up her arms and shoulders.
If, however, that is not enough, you could ask her to cover up the tattoos with make-up but that will depend on 1) how comfortable she is with covering them up and 2) if it is even possible.
You’re going to have to compromise, either in the style of dress or choosing between having your sister in the bridal party and not.
The fact is, she is your sister and, while I don’t know how close you both are, the fact is she must be special enough for you to want her in your bridal party. In that case, does it really matter if her tatts show? She is how she is and, given how many of them she has probably means they form part of her identity.
So, asking her to cover them up, while completely reasonable from your point of view as a bride-to-be, may be seen by her as you wanting her to hide something that is very important to her.
Dropping her from the bridal party is the most drastic of steps and should only be done if you’ve truly weighed up the potential damage it may do to your relationship. That may be worse than a few (well, maybe more than a few from the sounds of it) tattoos in your wedding photos!