[RPG] Does a monk with Unarmored movement still have to make strength checks to “run”/”climb” a wall

Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall

 x
 x
 x
 x
 x
 x
 x
 x
 x
 x

Now let's look at the veritcal space next to the wall

 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Finally we need to have the flat part on top of the wall, and the space above that

  yyyyyyyyy
 yxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).

  yyyyyyyyy
 bxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 ax

If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).

The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.

This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.

45' wall climbs (and lower) are safe though.

TL;DR: Yes, he can attack if he has enough Movement to make the climb and does not have to use two interactions to both stow and draw a weapon.

Walls can be covered under "special types of movement." (PHB, p. 182). The rules-as-written allows that attack if he has enough movement left to climb that distance, since you can generally move and attack in the same turn.

Climbing, Swimming, and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the DM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. (Basic Rules, p. 64 / PHB. p. 182)

If he had at least 20' of movement left, it fits RAW that he'd complete the move up the 10' wall and then attack, providing the character succeeded in the Strength (Athletics) Check you required. Since you have already determined that the wall isn't that hard to climb, there seems no reason not to allow the attack. If you had ruled it difficult terrain, he'd have needed 30' of movement to make the climb and then attack.

Other Activity on Your Turn:
[…] You can interact with one object or feature of the environment during either your move or your action. (PHB, p. 190, Basic Rules, p. 70)

What you may wish to rule on is whether or not he must

1. stow the weapon (interaction)
2. climb the wall (movement)

3. and then draw the weapon (interaction)

   If he needs two interactions (only one interaction is free) that would most often preclude the attack. (In some cases, a bonus action could still allow it.)
   You state that this wall is not hard to climb. You may or may not allow him to climb this wall with a weapon in hand. You could increase the DC of the climb if he attempts it with weapon in hand. As that wasn't specified in the question, I'm can't suggest a ruling either way.

Your Turn
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. Your speed—sometimes called your walking speed—is noted on your character sheet. (PHB, p. 189; From Basic Rules, p. 69)

Movement and Position

You can use as much or as little of your speed as you like on your turn, following the rules here. Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you’re moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving. (PHB, p. 190; Basic Rules, p. 70)

If the character is a Rogue with Second-Story Work (Thief Archetype) then he'd not need all 20 feet of move, just the 10 feet.

Second-Story Work
When you choose this archetype at 3rd level, you gain the ability to climb faster than normal; climbing no longer costs you extra movement. (PHB p. 97; Basic Rules, p. 28):