Egalitarianism proposes that everyone should have same access to political and civil rights, but does that mean include equal opportunity?
All political terms are open to a variety of meanings. Suppose we take egalitarianism in this sense :
The egalitarianism I have in mind has as its ideal a condition of equal well- being for all persons at the highest possible level of well-being, i.e. maximum equal well-being. Egalitarianism holds that society should be arranged so as to promote and maintain this state. (Bruce M. Landesman, 'Egalitarianism', Canadian Journal of Philosophy, Vol. 13, No. 1 (Mar., 1983), pp. 27-56: 27.)
This is wider than your definition but I think it is closer to the general sense of 'egalitarianism' in political philosophy; I don't see egalitarianism as restricted to political and civil rights. What about economic equality ?
Equality of opportunity
Equality of opportunity means that (at least in theory) everyone starts from the same base point, with no-one enjoying initial advantages. However the logic of equality of opportunity is that it is opportunity to achieve advantaged positions. We can introduce safety-net rules so that the losers under equality of opportunity don't fall below a certain minimum condition. We can also set upper limits to the permissible inequalities not to exceed certain limits.
But there is no logical or empirical connection between equality of opportunity as the initial state and the end-state of equal well-being for all persons