Given two disjoint circles $S_1$ and $S_2$, and a point $X$ external to both of them, is it possible to find the center of a circle that passes through $X$ and touches $S_1$ and $S_2$ tangentially, using only a straightedge. If yes, how could it be done?
I would also much appreciate any theorems that may help in this problem.
I don't think you can do this using only a straightedge. You will probably also need a compass, or a piece of string.
What you will need to do is draw two hyperbolae. I will assume you are sufficiently comfortable in working with hyperbolae. Moreover, I will not explicitely go over the construction of a hyperbola. This is well documented, for instance on Wikipedia or in this text (look at section 2.3 in particular to see a construction method which doesn't require a compass).
In the picture below I've drawn the two disjoint circles $S_1$ (radius $a$, centre $c_1$) and $S_2$ (radius $b$, centre $c_2$) and a point $X$.
The $\color{blue}{\text{blue}}$ hyperbola $P$ is the the set of centres of circles that are tangent to both $S_1$ and $S_2$. Note that the two branches of this hyperbola can be characterised as: $$P_\mathrm{ex}=\left\{\text{centres of circles $C$}\mid \text {$S_1$ and $S_2$ touch $C$ externally}\right\}\\ P_\mathrm{in}=\left\{\text{centres of circles $C$}\mid \text{$S_1$ and $S_2$ touch $C$ internally}\right\}.$$
The $\color{red}{\text{red}}$ hyperbola $Q$ is the the set of centres of circles that are tangent to $S_2$ and pass through $X$. The branches of this hyperbola can be characterised as: $$Q_\mathrm{ex}=\left\{\text{centres of circles $C$}\mid \text {$C$ passes through $X$ and touches $S_2$ externally}\right\}\\ Q_\mathrm{in}=\left\{\text{centres of circles $C$}\mid \text{$C$ passes through $X$ and touches $S_2$ internally}\right\}.$$
These are the hyperbolae you would need to custruct (at least partially).
By definition of our hyperbolae the intersection points $c_\mathrm{ex}:P_\mathrm{ex}\cap Q_\mathrm{ex}$ and $c_\mathrm{in}: P_\mathrm{in}\cap Q_\mathrm{in}$ are centres of the only two circles that pass through $X$ and are tangent to both $S_1$ and $S_2$. Where of course, $S_1$ and $S_2$ are externally tangent to the circle with centre $c_\mathrm{ex}$ and internally tangent to the circle with centre $c_\mathrm{in}$.