The necessary condition pointed out by Jens Reinhold is also sufficient: any torsion class $x = [M] \in \Omega^{SO}_d$ admits a representative where $M$ is a rational homology sphere.
EDIT: This is Theorem 8.3 in $\Lambda$-spheres by Barge, Lannes, Latour, and Vogel. They also calculate the group of rational homology spheres up to rational h-cobordism, and more. I'll leave my argument below:
To prove this, we first dispense with low-dimensional cases: in any dimension $d < 5$ the only torsion class is $0 = [S^d]$. The high dimensional case follows from Claims 1 and 2 below.
I'll write $MX$ for the Thom spectrum of a map $X \to BO$ and $\Omega^X_d \cong \pi_d(MX)$ for the bordism group of smooth $d$-manifolds equipped with $X$-structure. Representatives are smooth closed $d$-manifolds $M$ with some extra structure, which includes a continuous map $f: M \to X$.
Claim 1: if $d \geq 5$ and $X$ is simply connected and rationally $\lfloor d/2 \rfloor$-connected, then any class in $\Omega^X_d$ admits a representative where $M$ is a rational homology sphere.
Claim 2: There exists a simply connected space $X$ such that $\widetilde{H}_*(X;\mathbb{Z}[\frac12]) = 0$, and map $X \to BSO$ such that the image of the induced map $\Omega^X_d = \pi_d(MX) \to \pi_d(MSO) = \Omega_d^{SO}$ is precisely the torsion subgroup, for $d > 0$.
Proof of Claim 1: Starting from an arbitrary class in $\Omega^X_d$ we can use surgery to improve the representative. Since $X$ is simply connected and $d > 3$ we can use connected sum and then surgery on embeddings $S^1 \times D^{d-1} \hookrightarrow M$ to make $M$ simply connected. Slightly better, such surgeries can be used to make the map $M \to X$ be 2-connected, meaning that its homotopy fibers are simply connected. From now on we need not worry about basepoints and will write $\pi_{k+1}(X,M) = \pi_k(\mathrm{hofib}(M \to X))$. These are abelian groups for all $k$.
If there exists a $k < \lfloor d/2\rfloor$ with $\widetilde{H}_k(M;\mathbb{Q}) \neq 0$ we can choose $\lambda \in H_k(M;\mathbb{Q})$ and $\mu \in H_{d-k}(M;\mathbb{Q})$ with intersection number $\lambda \cdot \mu \neq 0$. If $d = 2k$ for even $k$ we can additionally assume $\lambda \cdot \lambda = 0$, since the signature of $M$ vanishes. The rational Hurewicz theorem implies that $\pi_k(M) \otimes \mathbb{Q} \to H_k(M;\mathbb{Q})$ is an isomorphism, and the long exact sequence implies that $\pi_{k+1}(X,M) \otimes \mathbb{Q} \to \pi_k(M)\otimes\mathbb{Q}$ is surjective. After replacing $\lambda$ by a non-zero multiple, we may therefore assume that it admits a lift to $\pi_{k+1}(X,M)$. Such an element can be represented by an embedding $j: S^k \times D^{d-k} \hookrightarrow M$, together with a null homotopy of the composition of $j$ with $M \to X$. In the case $k < d/2$ this follows from Smale-Hirsh theory, in the case $d = 2k$ we must also use $\lambda \cdot \lambda = 0$ to cancel any self-intersections. (Actually there could also be obstructions to this in the case $d=2k$ for odd $k$, but those obstructions vanish after multiplying $\lambda$ by 2.) The embedding and the nullhomotopy gives the necessary data to perform surgery on $M$ and to promote the surgered manifold to a representative for the same class in $\Omega^X_d$.
Performing the surgery gives a new manifold $M'$ where $H_k(M';\mathbb{Q})$ has strictly smaller dimension than $H_k(M;\mathbb{Q})$ and $\widetilde{H}_*(M';\mathbb{Q}) = 0$ for $* < k$. This is seen in the same way as in Kervaire-Milnor. The case $d > 2k+1$ is easy, similar to their Lemma 5.2. In the case $d = 2k+1$ the diagram on page 515 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_k(M')$. In the case $d = 2k$ the diagram on page 527 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_{k-1}(M')$.
In finitely many steps we arrive at a representative where $\widetilde{H}_k(M;\mathbb{Q}) = 0$ for all $k \leq \lfloor d/2\rfloor$. Poincaré duality then implies that $H_*(M;\mathbb{Q}) \cong H_*(S^d;\mathbb{Q})$. $\Box$.
Proof of Claim 2: Finiteness of the stable homotopy groups of spheres implies that $\pi_d(MX)$ is a torsion group for $d > 0$ for any such $X$. Therefore we can never hit more than the torsion in $\pi_d(MSO)$, all of which is exponent 2 by Wall's theorem. The difficult part is to construct an $X$ where all torsion is hit.
The non-trivial based map $S^1 \to BO$ factors through $\mathbb{R} P^\infty \to BO$, whose image in mod 2 homology generates the Pontryagin ring $H_*(BO;\mathbb{F}_2)$. We can freely extend to double loop maps
$$\Omega^2 S^3 \to \Omega^2 \Sigma^2 \mathbb{R}P^\infty \to BO$$
where the second map then induces a surjection on mod 2 homology. Both $\Omega^2 \Sigma^2 \mathbb{R}P^\infty$ and $BO$ split as $\mathbb{R} P^\infty$ times their 1-connected cover, so the induced map of 1-connected covers $\tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty) \to \tau_{\geq 2}(BO) = BSO$ also induces a surjection on mod 2 homology.
Now let $X = \tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty)$ with the map to $BSO$ constructed above. Take 1-connected covers of the double loop maps above, Thomify, 2-localize, and use the Hopkins-Mahowald theorem to get maps of $E_2$ ring spectra
$$H \mathbb{Z} _{(2)} \to MX_{(2)} \to MSO_{(2)}.$$
(See e.g. section 3 of this paper.)
We can view $MX_{(2)} \to MSO_{(2)}$ as a map of $H\mathbb{Z}_{(2)}$-module spectra, and hence $MX/2 \to MSO/2$ as a map of $H\mathbb{F}_2$-module spectra. The induced map $H_*(MX/2;\mathbb{F}_2) \to H_*(MSO/2;\mathbb{F}_2)$ is still surjective (it looks like two copies of $H_*(X;\mathbb{F}_2) \to H_*(BSO;\mathbb{F}_2))$, and inherits the structure of a module map over the mod 2 dual Steenrod algebra $\mathcal{A}^\vee = H_*(H\mathbb{F}_2;\mathbb{F}_2)$. Both modules are free, because any $H\mathbb{F}_2$-module spectrum splits as a wedge of suspensions of $H\mathbb{F}_2$. In fact the Hurewicz homomorphism $\pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2)$ induces an isomorphism
$$\mathcal{A}^\vee \otimes \pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2),$$
and similarly for $MSO$. Therefore the map $\pi_*(MX/2) \to \pi_*(MSO/2)$ may be identified with the map obtained by applying $\mathbb{F}_2 \otimes_{\mathcal{A}^\vee} (-)$ to the map on homology, showing that the induced map $\pi_*(MX/2) \to \pi_*(MSO/2)$ is also surjective. Now any 2-torsion class $x \in \pi_d(MSO)$ comes from $\pi_{d+1}(MSO/2)$, hence from $\pi_{d+1}(MX/2)$ and in particular from $\pi_d(MX)$. $\Box$
