$i.$ $S^1 \vee S^1$ can be embedded in a topological group
$ii.$ $S^1 \vee S^1$ can be covered by a topological group
I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which is a topological group under addition.
Not sure about $ii$ though. I know a covering map will induce an injective homomorphism on $\pi_1$ and that $\pi_1$ is abelian for topological groups. It appears that all of $S^1 \vee S^1$'s covering spaces have nonabelian fundamental groups, except for the universal cover, but I'm not seeing an obvious way of putting a group structure on that one.
ii is impossible. A topological group is locally homogeneous: for any two points $p,q$ there exist neighborhoods $U,V$ such that $U$ is homeomorphic to $V$. But no covering space of $S^1 \vee S^1$ has that property, because any covering space is a graph having a vertex of valence $4$, and the only locally homogeneous graph is a 1-manifold.