Can one prove without the Axiom of Choice that for every normed vector space $X$ there exist a nonzero continuous linear functional on $X$?
No, this is equivalent to the Hahn–Banach theorem already in the case of Banach spaces.
See in my write up: https://arxiv.org/abs/2010.15632
The answer is no. Let $\ell^\infty$ be the space of bounded sequences equipped with the $\sup$ norm, and $c_0$ its closed subspace of sequences convergent to zero. The quotient $\ell^\infty/c_0$ is then a Banach space when equipped with the quotient norm, and it is consistent that it does not have any nonzero continuous functionals - see a discussion in this answer.