People have interpreted your question in various ways, but I think the core of what you're asking is what happens if you jump straight ahead from a leading edge of a (uniformly) moving object, as opposed to jumping "backwards" at the other end.
So, within the context of this answer, a platform in uniform motion is assumed, and no other effects are considered.
Even before you jump, you already have Earth's velocity, by virtue of being bound to it (you are held by gravity, and you are moving together with the Earth).
So when you jump, you're not "suspended" in space, stationary with respect to the Sun. The Earth doesn't "catch up to you". When you jump off of the "leading edge", you just add a bit of velocity on top of what you already have, and on the other side, you just subtract a tiny portion of that velocity. Assuming (approximately) uniform motion of the Earth for the duration of the experiment, the motion is such that this exactly cancels out, and the result is the same as if the Earth was stationary to begin with. That's just the classical (Galilean) relativity of motion.
Earth's orbital speed is about 107000 km/h (67000 mph). If the Earth suddenly stopped with respect to the Sun, on the leading edge, it would look as if you jumped incredibly high on one side, and on the other, you'd get squashed.
This is analogous to a scenario where you're standing on a train, with the train moving in a straight line with constant velocity. You've already accelerated together with the train, so, to a stationary observer, you already have the same velocity as the train.
If you jump up, you land in the same place, but to an observer standing next to the tracks, you travel the same horizontal distance as the train. If you jump forward, for you it looks like you've traveled some small distance forward, but to a stationary observer, you traveled just slightly faster then the train itself.
If you jump backwards, it's similar. To you, it's a small distance. To a stationary observer, your speed was slightly lower than that of the train.
It's only when the train starts to accelerate (or decelerate, or turn) that you notice any difference, compared to the train standing still.
P.S. The reasoning behind my interpretation of the question:
I looks to me like the situation the OP is trying to set up is one where the person that is jumping is pushing off in the direction tangential to the orbit (either in the direction of Earth's orbital motion, or in the opposite direction). It seems like the OP is ignoring velocity due to rotation. E.g. The person is standing either on the leading or the retreating edge of the Earth's sphere, with the Sun positioned somewhere to their side (they are standing on the shadow terminator line).