The answers to date already say "no", but I wanted to point out some other issues with your logic. The final result is that RAW would allow you to do everything except let the horse take multiple turns in a round to move unlimited distances, but not quite the way you stated it.
That said, you should be able to move the horse indefinitely by just shoving it, though you will need high-level NPCs (around level 24), since they need a Strength (Athletics) skill of at least 15. And your rider will need a Dexterity modifier of at least +9 if you want to move horse and rider.
Of course, the biggest flaw in this whole thing is that you'd need to get a bazillion NPCs standing around waiting for you and your horse. A level 24 player or NPC could easily afford to just hire a level 13 wizard to teleport them to their destination in much less time than it would take all your NPCs to hike to their destinations.
Attack rolls won't get you anywhere near light speed, let alone infinity
If you were relying on attack rolls, you'd fail every 20th time, on average, even with +200 attack rolls vs an unarmored target.
If the d20 roll for an attack is a 1, the attack misses regardless of any modifiers or the target’s AC. (D&D basic rules, page 73)
To hit the speed of light, you'd have to make it through about 49 million npcs in one round. The odds of that are right at zero. (About 1 in 10^million ish. That's googol^(10000), or a 1 followed by a million zeros.)
Doing it with shove actions
However, the "shove" action isn't really an attack. It uses an attack action, and is considered "a special melee attack", but you actually make a contested Strength (Athletics) check against the target's Strength (Athletics) or Dexterity (Acrobatics), with the target choosing which.
The target of your shove must be no more than one size larger than you, and it must be within your reach. You make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics)
check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you. (D&D basic rules, page 74)
I'm not seeing anything that says it automatically knocks the person off the mount in the basic rules, even if the shove succeeds. Assuming there's something in the full PHB that makes this clear, you could modify the question to work better.
If each NPC has +20 Strength (Athletics) and +0 Dexterity (Acrobatics), then the NPC currently on the horse chooses to contest the shove with Acrobatics rather than Athletics, and the best roll they could get is natural 20 + 0 = 20. The NPC on the ground would have a worse case roll of natural 1 + 20 = 21, which always succeeds, since opposing skill checks have no automatic failure.
Problems with falling off via a shove
Now, if the full PHB doesn't say the mounted guy automatically falls off if he's been successfully shoved, there are three ways I can think of to shove someone off a mount.
First, the shove action can push a target 5 feet away from you. Since horses are generally not 5 feet wide, that means you're no longer on the mount. Since there's no explicit saving throw when the rider is moved against their will, it seems logical this an automatic dismount. However, it's not explicitly stated this way, at least in the basic rules.
Second, the shove action could be used against either the horse or the rider to knock it prone. Knocking the horse prone would screw with its movement (from D&D basic rules page 70, it requires half your movement speed to stand up), so we'd want to knock the rider prone. At this point, the rider needs to make a Dexterity check at DC 10 to avoid falling off. Since the lowest modifier you can get is -5 for an ability score of 1 (D&D basic rules page 57), the rider would succeed with a roll of 15 or higher, or 30% of the time. This is even worse than the attack roll scenario.
Third, the shove action could be used against the horse to move it 5 feet, which would trigger the same Dexterity check at DC 10. The cool part of this is you can hypothetical get an extra 5 feet per round if you shove the horse as it's passing you, moving it 5 feet in front of you, then mounting it since it's within 5 feet of you. But it's still not a good scenario.
Once during your move, you can mount a creature that
is within 5 feet of you or dismount.
If an effect moves your mount against its will while you’re on it, you must succeed on a DC 10 Dexterity saving throw or fall off the mount, landing prone in a space within 5 feet of it. If you’re knocked prone while
mounted, you must make the same saving throw.
If your mount is knocked prone, you can use your reaction to dismount it as it falls and land on your feet. Otherwise, you are dismounted and fall prone in a space within 5 feet it. (D&D basic rules, page 76)
Deliberately falling prone
Your intent to deliberately fall prone while on the mount might be construed as being "knocked prone", requiring each NPC to pass their subsequent Dexterity check at DC 10 to stay on. You could either just not do this, since the skill check can be guaranteed anyways, or you could ensure your riders have at least a +9 Dexterity modifier (28-29 score).
If each NPC has a +9 Dexterity modifier, it's Dexterity (Acrobatics) is necessarily at least 9, so they'd need a Strength (Athletics) of 29 or more.
Doing away with all the shoving
I'm not sure this is the intended interpretation. However, it seems that you can mount once per turn and dismount once per turn (the wording could also be interpreted as saying you can either mount or dismount in a turn, but not both). Since each action takes half your movement speed, each of your NPCs could simply mount the horse at the start of their turn, then dismount the horse at the end of their turn, allowing the next NPC to mount without shoving anyone.
A better way to move the horse indefinitely.
Ok, so the other answers noted that the horse can't take multiple turns and keep running. But nothing says we can't keep shoving the horse.
Instead of NPCs 120 feet apart, put them 5 feet apart. Assume the NPCs and the horse all allow each other to invade their protected area.
Next, assume the NPCs have a Strength (Athletics) skill that's at least 20 higher than one or both of the horse's Strength (Athletics) or Dexterity (Acrobatics) skills.
Now, the first NPC is directly behind the horse and shoves the horse forward 5 feet, just past the second NPC. The second NPC shoves the horse another 5 feet, etc. If we rule that you can't literally shove the horse through the other NPC, you could have each NPC stand to the side of the horse's path, then step behind the horse on their turn before shoving. Regardless, the horse moves 5 feet per NPC turn.
If you want, you can have a rider with a Dexterity modifier of +9 or more (to avoid falling off each time the horse is shoved). Then both horse and rider can move indefinitely by being shoved across the planes.
(You can shove something up to one size larger than you. I'm assuming horses are Large size. If not, your NPCs would need to be scaled accordingly.)
A note on skill levels
I've been playing through Icewind Dale and Knights of the Old Republic, which are based on earlier editions of D&D where you increment your skills each level. I'd forgotten that 5e has a baked-in proficiency bonus that's related to your character level rather than something you advance manually, which makes it much harder to get numbers like +20 on your checks.
That said, it should still be possible. First, assume the horse has a Dexterity of 1, giving it a -5 modifier. Now, the NPC "only" needs a +15 modifier to always beat it. At level 20, there's a +6 proficiency bonus. A Strength score of 28-29 gives an ability modifier of +9, for a total of +15.
I know you can get a Strength of 20 through leveling. I'm guessing there's some magic item or another that grants +8 to Strength. If not, find a race that does have 28 Strength for your NPCs. Additionally, my understanding is you can earn 2 Strength per level past 20 if your DM allows it. Even without extended proficiency bonuses, a level 24 NPC could have the required 28 Strength naturally.
Similarly, the rider needs a +9 Dexterity modifier to always beat the DC 10 check to avoid falling off the horse, which requires the same kind of solution as getting a +9 Strength modifier.
The problem of practicality
Sure, RAW might allow a bazillion and one NPCs to dutifully line up so we can move our horse (and/or rider) a million miles in 6 seconds flat.
But it's rather doubtful your DM is actually going to allow that. Since RAW allow the DM to govern the NPCs in a manner that makes sense, RAW allows the DM to automatically prevent your solution from ever starting to take place, let alone work.
The reason it doesn't matter
Ok, so we have a bazillion and one NPCs dutifully lined up to move our horse (and/or rider) a million miles in 6 seconds flat. We broke the game!
This is D&D, which is a fantasy game. In this fantasy world, is a cool little thing called the "Teleport" spell.
Casting Time: 1 action
Range: 10 feet
This spell instantly transports you and up to eight willing creatures of your choice that you can see within range, or a single object that you can see within range, to a destination you select. If you target an object, it must be able to fit entirely inside a 10-foot cube, and it can’t be held or carried by an unwilling creature.
The destination you choose must be known to you, and it must be on the same plane of existence as you. Your familiarity with the destination determines whether you arrive there successfully. The DM rolls d100 and consults the table. (D&D basic rules, page 102)
Teleport is a 7th level spell with only a verbal component. The only thing you need is a 13th level Wizard and you can move as far as you like within the same 6-second round. Consulting "the table", we can guarantee exact teleportation with no mishaps as long as you either have a sigil sequence to a permanent teleportation circle, or just an object from the destination taken from there within the last six months.