Nope. You don't need FTL.
Though you do need to move your thinking about warfare back a couple of centuries. And you need some seriously good environmental systems. Oh, and ridiculous engines. Oh, and some major production capacity.
Firstly: You need to stop thinking 'fleet' and start thinking 'armada'. Warfare in the age of tall ships is probably the closest analogue that we have here, when battles lasted weeks and voyages from country to country measured in months. Journeys to the enemy for you will take years, at best, so you need to take pretty much everything that you need to win the first battle, establish a beachhead and hold it until you can establish resupply. Given the timescales involved you'll need to have a trail of resupply missions following the initial hammerhead of your armada, and if the warfare is prolonged then each side will have vast amounts of resources doing nothing but shuttling back and forth in the space between stars.
Which brings me to resources and production. This is a major logistical nightmare, and your enemy has an insane home field advantage. Not only that, but you can't really call for help. You need to take as much production capacity as you possibly can with you. While certain things (people, high tech or sensitive equipment) must be sent from home en-masse, you'll basically want to be able to source basics like ammo, fuel and potables (air, food, water) when you get to the enemy star system.
Which brings me to recycling. This is where we need to get a bit soft-sciencey, or assume that your two sides really, really want to engage in this war. Basically the issue is that your ships will be leaking all the way to the enemy, regardless of how well you build them, and any losses on the way cannot be recouped, so either you need some super environmental systems, or you need to carry enough resources (air, mostly) with you that you can absorb any possible losses and not care.
Which brings me to engines. Unless you're planning on populating your warships with generations of some sort of engineered warrior culture (which is a really cool idea, but may be impractical) you need some serious propulsion. You're going to have to accelerate like hell until you get halfway, then flip and slow down. Doing this with an entire army, mobile refineries and mines, all your resources for the trip and initial occupation plus all the materials of war is going to take some serious, serious engines. When you take into account the tyranny of the rocket equation this only gets worse. You'll need something like Project Orion if you want to be hard science, but probably you'll need to go for something more soft-sci-fi.
Although taking all of this to its logical conclusion contested areas will have armadas of ships propelled by nukes filled with second generation warriors and military hardware being followed by endless streams of ships propelled by nukes filled with second generation merchants and material goods.
Or, to put it another way, space Constantinople.
EDIT: I Just noticed that the OP's question actually says intra system wars, not inter- system wars. This changes my answer a little, though only by relaxing the requirements somewhat as the travel times now go from decades to months or even weeks, and communication times now measure in minutes rather than years. This makes resupply easier and you no longer have to worry so much about environmental tolerances as you can probably set up regular resupply stations orbiting the star without too much expenditure. It also means that you don't have to have such badass engines. Indeed, if you really want to you could have solar sail propelled armadas.
Which really takes us back to the age of the tall ship!