As a bit of background, my players befriended a kobold and gave him a javelin (instead of the dagger a kobold usually have, according to his stat block). They later also gave him a leather armor for his protection.
Does the kobold have proficiency for the javelin and the leather armor ?
Or the general question: Do monsters have proficiency with all weapons and armors ?
If not, how can I know what they have proficiency with ?
It's the DM'S call.
From the Monster Manual, pg 9:
Assume that a creature is proficient with its armor, weapons, and tools. If you swap them out, you decide whether the creature is proficient with its new equipment.
That said, common sense goes a long way. Your kobold, if he's in a conventional setting, comes from a small to medium-sized tribe and would know how to use weapons that a tribal hunter would know- spears, javelins and the like. If his tribe is warlike he might know a few martial weapons; if his tribe works for a greater power he knows what he was trained to know. The same goes for armor.
If he has class levels, the kobold would of course have those proficiencies available to him as well.
Where 3.5 tended to err on spelling out everything, 5e tends to err on the side of leaving it up to the DM. Both have their pros and cons, but the bottom line for tweaking monsters in 5e is, "who are we to argue with you? It's your world, after all." (MM, p4)
However, if what you want is some DM advice, "do whatever you want" is not always very helpful, so here are some suggestions.
When you find that such a creature suddenly becomes more important than you expected them to, such as when your PCs decided to befriend a Kobold instead of killing it (good for them!) you suddenly need to flesh them out a bit, and their proficiencies is only one of the things you will have to figure out.
A good rule of thumb for assigning proficiencies is to pick the character class that seems most logical for the monster's race based on their entry in the Monster Manual, and adopt some or all of those proficiencies. In the case of a Kobold, you have a creature with Dex as their only good stat and that likes to set traps and attack in groups. Rogue sounds like the best fit to me! So unless this Kobold is especially unique, I would assume that it has many of the same proficiencies as a Rogue, even without giving it any actual levels.
Another challenge that is presented when the characters unexpectedly decide to talk to a monster instead of fight it is that now you have to role-play the monster differently than you expected. When you make an NPC that you know will be important, you build them with an idea of what sort of person they are, but if you did that with every individual monster in your game world you would be so bogged down in prep you would never be ready for game night!
I suggest having a small list of monster names and generalized motivations that you can slap right onto that one-in-a-thousand monsters that suddenly becomes important enough to warrant a personality. It is important to keep things general, so that a motivation that works for an orc also works for a kobold, etc. that way you can recycle your prep so that it won't get wasted even if it doesn't get used for a while. For these types of situations all you will probably need is a starting point and you can improvise the rest and dig deeper if the creature continues to be important. If the PCs want to keep them around long term, now they are a full-fledged NPC and warrant even more fleshing out.
