No. Getting to the Sun requires scrubbing off nearly all the orbital speed of the Earth (30km/s), or the payload just orbits at a smaller radius.
The delta v to the Moon is something like 7 times less than to the Sun (3 vs 20ish).
The delta v need of the Parker Space Probe, which still "only" gets to 8.5 solar radii is so high it will use 7 Venus assists to get it close enough (it can't do big Jupiter assists because the solar panels it would need at Jupiter wouldn't be able to fit behind the sun shield at perihelion, and they didn't want to give it an RTG because they're saving plutonium for future missions).
You can eventually hit the Sun with enough Venus assists and Earth assists, for a total delta v of under 4 km/s, but it'll take a very, very long time, and your nuclear waste will be doing Earth flybys until finally it hits the Sun. Also it might be tricky to get the ball of sun-melted radioactive slag to do accurate assists after the sun melts it on the last few close encounters.
There's a few major issues with disposing of nuclear waste into the sun. First and foremost is that if the rocket explodes you'll contaminate a huge radius with dirty bomb like material. Probably not great.
Further, if you do decide to launch it, it takes significantly more energy to shoot it out of the solar system than towards the sun. [1]
It's a misconception that shooting things to the Sun is easy, it's actually extremely difficult due to the fact that when you launch off of the Earth, after escaping Earth's gravity you're left with ~30km/s of leftover momentum that you have to do something with in order to not wind up in an orbit around the Sun.
The momentum you inherit from Earth isn't in the direction of the sun, but roughly perpendicular to that direction.
It's similar to a car going down the freeway at high speed that needs to make an immediate 90 degree right turn. It can't do that without slowing down first, if it did, it would slide sideways off the sideroad. It's harder in space because the car is going 30 km/s and there's no friction that slows you down.
Imagine sitting on a huge carousel spinning fast, like 100 times the speed of sound fast. Now imagine jumping off and trying to get to the center of it.
Yes, gravity is holding us back from yeeting of into space but to get closer to the sun you need to slow down. A lot. It takes way less rocket fuel to speed up enough to leave the solar system than to get to the sun.
Actually nope. It takes more energy to slow yourself down enough to crash into the sun, starting from the earth, than it does to hop on over to the moon.
Definitely a stupid question, because I actually have no idea how they work: can a solar sail be "tacked" so as to decelerate towards the sun, like a sailboat sailing into the wind?
(Setting aside the risky problem of getting this waste off earth and into space to begin with)
Tacking works because boats have a second "sail" interacting with a second fluid moving differently than the wind (the keel is in the water). There's no second thing to push against in space.
The delta v to the Moon is something like 7 times less than to the Sun (3 vs 20ish).
The delta v need of the Parker Space Probe, which still "only" gets to 8.5 solar radii is so high it will use 7 Venus assists to get it close enough (it can't do big Jupiter assists because the solar panels it would need at Jupiter wouldn't be able to fit behind the sun shield at perihelion, and they didn't want to give it an RTG because they're saving plutonium for future missions).
You can eventually hit the Sun with enough Venus assists and Earth assists, for a total delta v of under 4 km/s, but it'll take a very, very long time, and your nuclear waste will be doing Earth flybys until finally it hits the Sun. Also it might be tricky to get the ball of sun-melted radioactive slag to do accurate assists after the sun melts it on the last few close encounters.