You can force your way in with tumble/overrun if there is space
Before we dive in, can a creature from the inside even try to block the entrance?
The the entrance size is not explicitly defined, but as you can look out through it as if through a 3-foot-by-5-foot window, I think it is save to assume that it is 3x5 feet large.
A medium or small creature can position themselves on the space on inside of the entrance. It does not matter if the entrance is oriented towards the bottom, or sideways, as creatures control a 5x5 foot space they occupy, and there would at least be 2 feet of ledge to stand on in that space if the entrance is only 3 feet wide. A creatur controls the space it is occupying. So yes, they can block the entrance by controllng the space on the inside.
You cannot enter if there is no space
Rope Trick states:
The space can hold as many as eight Medium or smaller creatures.
If the Rope trick is already filled to the brim with creatures, that is, all the space is taken up, there is no unoccupied space, and since the rules (p. 191 PHB) state that "Whether a creature is a friend or an enemy, you can’t willingly end your move in its space.", you cannot move in, as you would need to end your move on another creature's space.
Some people also interpret this sentence from Rope Trick to mean that only Medium or Small creatures can even enter the space. That is clearly not the designer's intention, so I will also consider how it works for Tiny or Large creatures, and you can ignore those sections if you feel that they cannot, but I want to provide a complete answer also for those that allow other sizes of creatures to enter.
You cannot enter without action options
The key sentence is this one:
Attacks and spells can't cross through the entrance into or out of the extradimensional space, but those inside can see out of it as if through a 3-foot-by-5-foot window centered on the rope.
First, you cannot shove the blocker out of the way. Shoving a Creature states (p. 195 PHB):
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.
Attacks cannot cross the entrance, and since shoving is an attack, so you cannot shove through the entrance.
Second, you cannot move through the blocker's space. Moving Around Other Creatures rules (p. 191 PHB) state:
you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.
So if the space is occupied by a medium or small creature, another medium or small creature cannot move through their space, as they are not two size categories larger or smaller than the creature blocking the entrance. By standard rules this means such creatures cannot force their way in.
If you allow other sizes to enter: A Tiny creature could move in, if the entry was blocked by Medium or larger creature. A Large creature could move in, if the entry was blocked by a Small or Tiny creature.
You can with action options
I'll consider "forcing" your way in as any means to enter the space against the will of the creature blocking the entrance. The DMG has these action options on page 272:
Overrun When a creature tries to move through a hostile creature's space, the mover can try to force its way through by overrunning the hostile creature. As an action or a bonus action, the mover makes a Strength (Athletics) check contested by the hostile creature's Strength (Athletics) check. (...) If the mover wins the contest, it can move through the
hostile creature's space once this turn.
Tumble A creature can try to tumble through a hostile creature's
space, ducking and weaving past the opponent. As an action or a bonus action, the tumbler makes a Dexterity (Acrobatics) check contested by the hostile creature's Dexterity (Acrobatics) check. If the tumbler wins the contest, it can move through the hostile creature's space once this turn.
These are not attacks, they are contested skill checks. Because they are neither attacks nor spells, they work across the entrance. Either one would allow you to move through the blocking creature's space and enter the Rope Trick, if you win the contest (and there is space left to move into).