Is the fixed point set of an involution on a topological manifold a submanifold?

Let $f:X\to X$ be a homeomorphism of a topological manifold with $f^2=\mathrm{id}$.

Is each connected component of $\{x\in X \mid f(x)=x\}$ a topological submanifold?

No, the connected components of the fixed points of $f$ need not be a manifold. I can give an example of a homeomorphism $f\colon \mathbb{R}^4\to\mathbb{R}^4$ with $f^2=\rm id$, whose set of fixed points is connected (in fact, contractible), but is not a manifold.

Let $X=B\times\mathbb{R}$, where $B$ is Bing's dogbone space, and define $f\colon X\to X$ by $f(b,r)=(b,-r)$. This is clearly a homeomorphism with $f^2=\rm id$ and $\lbrace x\in X\colon f(x)=x\rbrace=B\times\lbrace0\rbrace\cong B$. It is known that $X$ is homeomorphic to $\mathbb{R}^4$, but the dogbone space $B$ is not a topological manifold. Also, $B$ is contractible, as its product with $\mathbb{R}$ is contractible.

Note: Bing's construction of the dogbone space was published in May 1957¹, and the fact that its product with $\mathbb{R}$ is $\mathbb{R}^4$ was demonstrated in 1959². Eilenberg's list of open problems linked to by Willie Wong in the comment to the question dates back to April 1949³. So, it is entirely plausible that the problem was indeed open when Eilenberg published his list.

¹ A Decomposition of $E^3$ into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from $E^3$, R. H. Bing, Annals of Mathematics Second Series, Vol. 65, No. 3 (May, 1957), pp. 484-500. (Link)

² The Cartesian Product of a Certain Nonmanifold and a Line is $E^4$, R. H. Bing, Annals of Mathematics Second Series, Vol. 70, No. 3 (Nov., 1959), pp. 399-412. (Link)

³ On the Problems of Topology, Samuel Eilenberg, Annals of Mathematics Second Series, Vol. 50, No. 2 (Apr., 1949), pp. 247-260. (Link)