Oh, absolutely. The Inca Empire had no form of writing, but had massive armies. I'd give you a specific source, but honestly there's hundreds, easily googleable.

For Eurasians - it's a lot harder to tell. In order for us to be entirely certain, we have to know when the first army was in a region, and when writing reached that region. Without writing to record it, however, we don't know when there was an army!

Thus, the only really effective way to be sure was for another civilisation, with writing, to discover an army from a civilisation that had no writing. The literate group would then write about it. This clearly happened with the Spanish and the Incas, and debatably with the Mongols (but while they were themselves without writing, they had access to communities that had writing, making things a bit more muddled)

Here's another way of looking at this.

Civilisation is the granddaddy of this sort of computer game, where you start at the dawn of civilisation and work your way up the tech tree. The very first military unit in Civ is a 'warrior', represented as a guy with a club.

But what's the first thing you build in Civ? It's a city.

A city represents a number of abilities. There's specialisation - rather than everyone being farmers, one person does one thing, another does something else. There's the ability to organise relatively large groups of people, to make sure they follow the rules and get along. Usually this means a centralised authority to boss people around and enforce the rules.

Those are all capabilities that make armies possible. An army is a large group of people organised for the purpose of making war. Usually they're away from their homes for a period of time, and require some compensation or compulsion from their leaders to make them take part.

All of this requires the ability to organise and administer large groups of people, but it doesn't require writing. (Although writing can help immensely!)

One important caveat to this. Many computer games, like Civ, will give the impression of a standing army. That is a group of people permanently employed as soldiers, as in a modern military. Once recruited, that warrior unit of yours stays around until you dismiss it (sometimes hanging around for millenia...).

Throughout history (if I can risk a big generalisation) the majority of armies have been raised on a temporary basis, typically for a particular campaign. Campaigning was often seasonal, so armies were raised and dismissed with the seasons. (For example, it was often too cold in the winter, and people needed to be home to sow in the spring and harvest in the autumn, so the summer was campaign season...)

A standing army generally requires more sophistication, as a ruler needs substantial and reliable tax revenues to pay for it - and failing to pay your army is a dangerous thing. So, for example, in Europe, the Romans had a substantial army, but after the fall of the Western Empire, standing armies are incredibly rare until the end of the Middle Ages.

"The essential soldier remains the same. Whether he was handling a slingshot weapon on Hadrian's wall or whether he's in a main battle tank today, he is essentially the same." Gen. Sir John Hackett

There is practically no direct evidence regarding the political and military structure of the earliest civilizations, when various tribes in the Middle East were first learning how to grow crops and domesticate animals, and when the first villages began to grow into towns. But war must have already have been changing into a disciplined business with political and economic purposes that we would understand, for at least as early at 7000BCE.
Jericho has evidence of settlement back to 9000BCE I'm not 100% at what point they first build walls, but it predates writing and walls are a sure sign that you want to protect what you have from organized groups of armed men.