There is no difference between an object that is “at rest” (i.e. not moving) and an object that is moving at a constant speed of 100,000mph. If you were in a spaceship with no windows, be traveling at a constant speed with no acceleration, there would literally be no test you could perform to determine if you’re at rest or moving at a constant speed. That’s because motion and speed is all relative. There’s no absolute speed. Even though you may not realize it, whenever you talk about how fast something is moving, it’s always measured relative to something else (usually either yourself, or the Earth).
When it comes to the solar system, the planets’ orbits are mostly defined by the pull of the sun’s gravity. Our motion or non-motion relative to other stars and galaxies has absolutely no effect on the orbits of the planets in our solar system.
I think the reason that you intuitively think that our relative motion through the universe should cause an effect is because you’re used to that being the case on earth. If you tie a ball to a string and swing it around in a circular motion, you notice that the ball has various forces applied to it as a result of this motion. The string pulls on the ball, gravity continues to pull the ball down, air resistance resists the motion of the ball, etc. However, in space, these forces aren’t relevant, except gravity. And, gravity isn’t actually a force. Look up the Veritaseum video on YouTube about gravity not being a force.
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