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.