Assume a perfect solar system i.e. Earth revolves around the Sun in a circle of radius R, Sun being at the center. The locus of Earth is $x^2 + y^2 = R^2$
I have a moving coordinate system whose origin is always attached with Earth. Is it possible to derive the locus of the Sun as seen by Earth?
This feels like it may be too simple to be what you're looking for, but I'll go ahead and write it anyway, because sometimes we all stumble on stuff that in retrospect seem simple.
Let's say your orbit is parameterized by $\vec{r}=R\cos(\omega t)\hat{x} + R\sin(\omega t)\hat{y} $, I don't want to say it's the Earth relative to the Sun because we know that would be true only as an approximation. Let it be just some abstract circular trajectory, parameterized by $t$ and where the object is moving with angular frequency $\omega$ (you can also verify that $r_x^2+r_y^2=R^2$).
Now, apply the following coordinate transformation $\vec{r} \rightarrow \vec{r}^{\ \prime}$ defined by:
$$ x' = x-R\cos(\omega t), \ \ y' = y-R\sin(\omega t), $$
so that clearly in this coordinate system, the orbiting body always sits at the origin $\vec{r}^{\ \prime}=\vec{0}$.
Now, the position of the "Sun" (the center of the previous coordinate system) is found by simply putting $x=y=0$ and finding:
$$ x'=-R\cos(\omega t) , \ \ y' = -R\sin(\omega t), $$
which may also be written as $\vec{r}^{\ \prime} = -R\cos(\omega t)\hat{x}-R\sin(\omega t)\hat{y}$. If you don't like the negative radius, it may also be useful to note this can be written as:
$$ \vec{r}^{\ \prime} = R\cos(\pi + \omega t)\hat{x}+R\sin(\pi + \omega t)\hat{y}, $$
geometrically this means that as measured from the orbiting body, the center of rotation appears to be rotating in a circle that's centered on the body, with the same angular frequency, but with a relative phase of $\pi$. The source of this relative phase becomes very clear if you draw a little diagram, and see that the vector drawn from the center to the body is at all times equal and opposite to the vector drawn from the body towards the center.
In a non-rotating reference frame centered on the Earth, the position vector of the Sun is the opposite of the vector from the Sun to the Earth. Since the latter moves in a circle, so does the former.
In a reference frame centered on the Earth with one axis always pointing in the direction of the Sun and remaining axes aligned with the orbital plane, the Sun is stationary by definition. Realistically, though, it will appear to oscillate back and forth slightly due to the eccentricity of Earth's orbit.
In a reference frame centered on the Earth and rotating with it, the position vector of the Sun will move in a cone whose angle with Earth's rotational axis oscillates between $23.44^\circ$ and $-23.44^\circ$ over the course of a year due to Earth's axial tilt.
