Can a younger sister provide eggs for her sister to have a baby? Who is the mother if so?

Yes she can. The egss(donor) from the sister can be used for the other sister's pregnancy. The "mother" would be the designated sister who received the fertilized egg, unless she is a surrogate mother.Always consult with an attorney to make sure that "parental rights" are made clear from the outset with a legally binding agreement.
Define mother. If you are looking for an easy answer there is none, unless you exclude the genetic factors from the equation. It is hard to process the idea that someone who carries a baby to term is not the mother, but she is not truly the genetic mother, the donor sister is. It is likely the child would have a very similar gene profile, but not the same as if her "mother" were able to conceive her.