You are asking the following: Pick $x\in X$ and for $G=Homeo(X)$ consider the orbit map $o_x: G\to X$, $o_x(g)=g(x)$. Assuming that $o_x$ is surjective, when does it have a (continuous) section $s: X\to G$?
A general observation is that if $X$ is not contractible, then no section $s: X\to G$ is homotopic to a constant map.
Here is what happens in the case when $X$ is a closed connected oriented 2-dimensional manifold.
If $\chi(X)=0$, then $X\cong S^1\times S^1$ and, hence, is homeomorphic to a Lie group and hence, $o_x$ has a section.
If $\chi(X)=2$, then $G=Homeo(X)$ is homotopy-equivalent to $O(3)$, hence, $\pi_2(G)=0$. However, if there is a section $s: X\to G$, then $\pi_2(G)\ne 0$, proving that a section does not exist.
If $\chi(X)<0$ then $G$ is homotopy equivalent to ${\mathbb Z}$ (the identity component of $G$ is contractible). Hence, you cannot have a section $s: X\to G$.
I suspect that when $X$ is the $n$-dimensional sphere then a section $s: X\to G$ exists if and only if $n=1, 3, 7$, i.e. when $S^n$ has structure of a Lie group.
See
M.-E. Hamstorm, Homotopy groups of the space of homeomorphisms on a 2-manifold, Illinois J. Math. Volume 10, Issue 4 (1966), 563-573
for the details concerning homotopy types of the homeomorphism groups of compact surfaces. (The same conclusion applies when one works with the groups of diffeomorphisms, resp. PL homeomorphisms: Earle and Eells, 1966; resp. Scott, 1970.)
Edit: I just realized that for some reason, you are interested in the existence of a section which is continuous away from one point (I am not sure why though). This changes the answer. Namely, for $X=S^2$ (or sphere of any dimension for this matter), the map $\Theta$ exists, it is a rotation in the plane passing through $x, y$. On the other hand, if $\chi(X)<0$ the map $\Theta$ still does not exist since $X-\{x\}$ is not contractible and the argument in the case 3 still applies.