To first address the crux of the OP's curiosity:

The reason why the question popped in my mind is because Kylo Ren and Rey both were trying their best to let the First Order ship go and stop it from flying away respectively, as portrayed in the Star Wars: The Rise of Skywalker. They were using the Force to control objects. Even Darth Vader had that kind of power. Hence the doubt.

The ship that was stopped by Rey in Rise of Skywalker was a First Order transport, equipped with just four ion engines. In comparison, the ship Leia was on (i.e., the ship that had the Death Star plans) was the Tantive IV -- a CR90 Corellian Corvette with eleven turbine ion engines. Additionally, the First Order transport was well within that planet's gravitational pull, however, the Tantive IV was in outer space, unencumbered by gravity.

Given this, IMO, Vader probably didn't have the Force strength to stop a ship with that kind of power.

And then, additionally...

Darth Vader could have used his force to stop the ship or crush them probably.

Vader wanted to recover the Death Star plans, not simply destroy/kill the rebels that had it. If Vader had just destroyed the ship then, in theory, it would have been a lot more effort to sift through all the wreckage than to just assault and board the ship and then interrogate/torture the passengers until they surrendered the plans. In that regard, at least, Vader was successful (minus the whole droid escape pod thing, you know).

First, it's a movie where everything needs to be exciting, so that would rob us of the epic ending from a storytelling perspective. Otherwise here's a few ideas...

When trained in the Jedi temple, each Jedi has a specialization and focuses their skills in certain areas.. Darth maul focused solely on lightsaber fighting and was one of the best fighters, but barely had any knowledge of the force in terms of powers other than force push. Jedi healers heal soldiers during wars and sick citizens, but they'd be useless in a fight. Anakin had a more than basic knowledge of the force, such as force lifting people, choke(which is really just doing a force push on both sides of your neck), force push, and great dueling skills.. but he is not a master of the force like Yoda or Sidious is in terms of force powers.. so while he is strong, and we have seen Jedi such as Yoda lift a one man fighter out of a swamp, Vader probably doesn't have the force strength to pull an entire space shuttle out of orbit or maybe lacked concentration to do so(Yoda was alone and focused).

Vader had a 150 lb suit with a respirator and 3 robotic limbs, and pieces keeping him from dying.. the suit greatly diminishes the true strength of Anakin, as he would be a lot more powerful without it. Luke beat him with just a year of training from Yoda and a few years of self taught lessons between episodes 4 and 5, although it could be argued Anakin was battling his feelings as well.

The Disney trilogy in my opinion wasn't well written and overpowered the force on the characters..but she pulled a ship out of anger and used the dark side ,Rey is supposedly extremely strong and could even be a replacement for Sidious.