Does the bordism homology theory satisfy the weak equivalence axiom?

This answer is simply to write the details for my comment above. It amounts to doing a little more work with homotopy equivalences, so as to carry out essentially the argument you gave in your comment regarding the smooth case.

Assume that $f:X\to Y$ is a weak equivalence of topological spaces. We want to show that the map induced on topological bordism (as you define it) is a bijection.

Regarding the surjectivity, I will simply repeat the content of Tom Goodwillie's comment. First, we observe that any topological manifold has the homotopy type of a CW-complex (you can find references to proofs by Hanner in Milnor's "On spaces having the homotopy type of a CW-complex", as remarked in the comment of Greg Friedman). It follows that $f$ induces a bijection on homotopy classes of maps $f_\ast :[M,X]\to [M,Y]$. Therefore, any map $g:M\to Y$ is homotopic and thus cobordant to a map which lifts to $X$. This proves surjectivity.

To prove injectivity, we need to know that the inclusion of the boundary of a topological manifold is a closed Hurewicz cofibration, that is a NDR-pair. This holds since the boundary of a topological manifold is collared (see chapter 2 of Ferry's notes). With this fact at hand, we can show that for any topological manifold $M$, the pair $(M,\partial M)$ is homotopy equivalent rel $\partial M$ to a pair $(A,\partial M)$ such that the inclusion $\partial M\to A$ is a Serre cofibration.

Proof of claim: Simply factor the inclusion $\partial M\to M$ as a Serre cofibration followed by a weak equivalence $\partial M\to A\to M$. Since $\partial M$ is a topological manifold, it has the homotopy type of a CW-complex, and it follows that $A$ also has the homotopy type of a CW-complex. Then the map $A\to M$ is a weak equivalence of spaces with the homotopy type of CW-complexes, and thus a homotopy equivalence. The map $A\to M$ is therefore a homotopy equivalence rel $\partial M$, since both maps $\partial M\to A$ and $\partial M \to M$ are Hurewicz cofibrations.

We can now prove injectivity by carrying out an argument similar to the one you wrote in your comment. Given bordism classes $g_0:M_0\to X$, $g_1:M_1\to X$ in $X$ and a cobordism $g:M\to Y$ ($\partial M=M_0\coprod M_1$) between $f\circ g_0$ and $f\circ g_1$, we need to find a cobordism $g':M\to X$ between $g_0$ and $g_1$. For that purpose, we replace $(M,\partial M)$ with $(A,\partial M)$ as described above. Since $\partial M\to A$ is a Serre cofibration, we can find a lift up to homotopy $A\to X$ of the composite $A\to M\to Y$, which furthermore extends the map $(g_0,g_1):\partial M\to X$. Composing with the homotopy equivalence $M\to A$ rel $\partial M$, we get the desired map $g':M\to X$.