Does Jane Eyre fight for social equality? How does she remain acceptable within her time era?

Jane Eyre does not fight for social equality, if we understand that as joining a social movement. She doesn't, for example, become part of an organization fighting for women's equality, which would have been a radical move in her time period. Instead, she stays acceptable by adhering to the Christian moral values of women in her social class and dressing and behaving, on the whole, neatly and modestly. She looks and acts on the surface like the "Quaker" that Rochester compares her to. (Quakers were known for their quietness and for dressing in very sober colors, such as gray.) Jane rages against the restraints put on women, but it is largely an inward rather than an outward rebellion she expresses:

It is in vain to say human beings ought to be satisfied with tranquillity: they must have action; and they will make it if they cannot find it. Millions are condemned to a stiller doom than mine, and millions are in silent revolt against their lot. Nobody knows how many rebellions besides political rebellions ferment in the masses of life which people earth. Women are supposed to be very calm generally: but women feel just as men feel; they need exercise for their faculties, and a field for their efforts, as much as their brothers do ...

When Jane does outwardly rebel, it is against having her moral values compromised: for example, she runs off when Rochester wants to marry her while also being married to Bertha. It is a violation of Christian norms to engage in bigamy, and Jane won't allow herself to be compromised in that way. She frames her anger and rebellion individually, in terms of her own conscience rather than as a fight for larger social change (much as she might long for that). She primarily is true to herself and to her own integrity as a thinking, feeling, and caring human being. As she puts it:

I can live alone, if self-respect, and circumstances require me so to do. I need not sell my soul to buy bliss. I have an inward treasure born with me, which can keep me alive if all extraneous delights should be withheld, or offered only at a price I cannot afford to give.

It should be noted that Jane was criticized as a character when the novel first appeared because her strong emotions were considered unnatural in a woman. Nevertheless, she remained acceptable because she stayed within the confines of the Victorian home and family, even after running away, when she was rescued by upright people who turned out to be relations. In the end, she married and became an acceptable and devoted wife, even if her marriage was more equal than most.