The statement "a morphism is a surjection iff it is an epimorphism" holds in every topos, regardless of the law of excluded middle.

The precise proof depends on your notion of "surjection" (in a topos all reasonable internal notions of a surjection coincide --- in fact, due to the above statement, one may define a surjection as an epimorphism).

Perhaps the most obvious notion is: a morphism $s \colon A \rightarrow B$ is a surjection if whenever $b \in B$ then $\underset{a\in A}\exists s(a) = b$ in the internal logic of the category. If a category is regular, then such surjections coincide with covers. And covers are another obvious notion for surjections: a morphism $s \colon A \rightarrow B$ is a surjection if in the image-factorisation $A \rightarrow s[A] \rightarrow B$ the monomorphism $s[A] \rightarrow B$ is iso.

Since every topos is a balanced category, in every topos covers coincide with epimorphisms.

The statement "a morphism is a surjection iff it is an epimorphism" holds in every topos, regardless of the law of excluded middle.

The precise proof depends on your notion of "surjection" (in a topos all reasonable internal notions of a surjection coincide --- in fact, due to the above statement, one may define a surjection as an epimorphism).

Perhaps the most obvious notion is: a morphism $s \colon A \rightarrow B$ is a surjection if whenever $b \in B$ then $\underset{a\in A}\exists s(a) = b$ in the internal logic of the category. If a category is regular, then such surjections coincide with covers. And covers are another obvious notion for surjections: a morphism $s \colon A \rightarrow B$ is a surjection (i.e. cover) if in the image-factorisation $A \rightarrow s[A] \rightarrow B$ the monomorphism $s[A] \rightarrow B$ is iso.

Since every topos is a balanced category, in every topos covers coincide with epimorphisms.