In the military rank feature it says the soldier can "requisition simple equipment or a horse for temporary use". What qualifies as simple equipment, any simple weapon or something less complex than a catapult? Additionally, which part of the sentence is the "for temporary use referring to, the simple equipment and the horse or just the simple equipment?
This is going to be up to the DM, but if you are the DM asking what you should do, I would suggest that "temporary use" means "for as long as you need to complete a concrete objective task".
The temporary use is referring to both the horses and the equipment. You have quote the exact text wrongly, leading to the confusion:
your rank to exert influence over other soldiers and requisition simple equipment or horses for temporary use.
As a DM, I would require players to pay for any consumable or lost/damaged items.
The simple equipment, is referring to items found on the equipment list in chapter 5, and the listed weapons and armor.
This heavily depends on who your characters are, what they are doing, and where they are doing it.
Who: If you're characters are low level adventurers, even the best stocked company is not going to entrust them with high value gear. They'll probably get some riding horses, armor if they really need it and access to weapons.
What: If they just need it for travel, the local armory can probably handle that. But if they're going on a suicide mission for their own profit, they probably can't do that (At least not without some serious cajoling). If they are defending the local town from the oncoming hordes, the armory should be fully open to them.
Where: This is probably the most important factor. Not every small town armory is going to have a couple of suits of full plate around (and if they do, they might not fit your PC). Warhorses are likely to be even less common as they require resources and training that would be outside of the realms of possibility. A good guide for this would be to set a GP value for equipment available (the Lost Mines of Phandelver adventure uses 25 GP as the threshold for available items at their local store, do something similar for a small town armory).
As far as how long, a requisition is almost always for a specific task rather than a general thing. So your PCs should be required to bring it back, and in good working order, when their task is complete. Make it work like a library where they can't requisition other supplies and have to pay late fees if they do not bring things back within a certain period after the completion of their mission.
This depends on your setting. Many elements of D&D 5e backgrounds need the context of specific setting context in order to make a specific rulings.
For example the answer to your question is very different if you are playing a typical pseudo-medieval setting like Greyhawk versus a setting based on John Boorman's Excalibur.
In Greyhawk the answer is not it is not plausible that those with a solider background would have access to high value items like plate armor or even warhorses. In many Greyhawk realms, Plate Armor and warhorses are reserved for the use of nobles. It would be plausible for the players to grab a spear, short swords, provisions, chainmail, etc. Things that an ordinary veteran fighter would be equipped with.
In John Boorman's Excalibur in contrast plate armor is common and just about everybody wears it. The knights just have better looking more stylish armor. In this case it wold be perfectly plausible for a ordinary solider to have access to plate armor.
Another example is if your setting is more oriented towards the Renaissance or English Civil War period. There plate breastplates were possessed by the common solider.
If you have a reason to grant access to plate armor and it makes sense then go ahead and do it. Unlike prior editions D&D 5e is neutral to mundane equipment largely because Armor Class follows a absolute scale due to bounded accuracy. It will help but it is not the overwhelming advantage for low level character that it was in prior editions. Without plate, fighters are typically getting 17 to 19 AC in my company versus the maximum of 20 AC for plate + shield.
Also realize that there is no dexterity bonus in plate, that imposes disadvantage on stealth and requires a minimum strength of 15.