I'd say no, but the correct answer is, as usual: "It depends." Since factory loads are going to fall under a sort of law of averages, I wouldn't worry about it. The people who concern themselves with accuracy all hand-load their own ammunition and tune it to the rifle (I'll explain more in a minute), because they can attain better characteristics and consistency than the factory.

If you get a rifle chambered in 5.56, you're going to be absolutely fine regardless of the ammunition you buy. The reason for this is because 5.56 NATO is actually a higher pressure cartridge than .223 REM. I don't remember by how much without looking it up, but I believe it's on the order of 3-5K PSI. If you buy a rifle chambered in .223, there is a slight risk that it may not be designed to handle the increased pressure and you may see some issues related to increased wear and reduced accuracy. There are minor physical difference between the cases too, but IMO it's not as significant as the pressure differences. On the other hand, .223 chambered rifles may not ever have an issue with 5.56. It's all down to manufacturing variability and luck.

Now, the reason high-accuracy shooters hand load:

When you fire a cartridge, there is a shockwave that travels down the barrel and induces very slight flexing at the natural resonance frequency of that specific barrel. It's only slight, but it does affect accuracy. High-accuracy shooters tune their cartridges such that the bullet exits the barrel at the point where the barrel vibration reaches the center point of its sinusoid (well, the three dimensional equivalent since it vibrates in 3D space). You'll almost never attain this degree of accuracy with factory ammo, regardless of the type, since they aim for a middle-of-the-road load that's both safe and functions the action of the widest array of guns on the market. After all, you need enough pressure to impinge on the action or gas piston to cycle it, for example, but not so much the gun blows up--then it's a matter of cost reductions to get away with the least amount of materials while still retaining said functionality.

But, theory aside, it's not going to matter. The way I look at it for myself is that I don't shoot often enough to worry about super tight groups with tiny MOAs at long ranges, nor am I a competitive shooter. The gun's gonna shoot better than I ever could. In your case, I'd focus on familiarizing yourself with the gun. This will be difficult if you can't find ammo, but you do need to at least spend some time dry firing (snap caps help reduce wear--buy locally!), getting experience with the controls to build muscle memory, and if possible you'll want to put a couple hundred rounds through it. AR15s don't have much recoil, but you definitely want to get a feel for both how you and the gun react. Since you want it for defense, high accuracy shot placement isn't going to matter much in close quarters. Muscle memory, however, is everything. I really can't emphasize this point enough without either annoying the Hell out of you over something literally everyone interested in guns already knows (or expects) or sounding the part of a broken record. Or both.

If you have someone you trust to help teach you what you need to know, that would be ideal. Otherwise, ask the gun shop. They'll be able to answer all your questions on safety, familiarity, etc., since there will never be enough space for me to write an entire treatise here! And some things you just have to see in person.

I didn't intend to turn this into an essay, so I apologize for the length of this response. Some of this is also opinion and stream-of-consciousness writing, so if something isn't clear, please ask so that I may clarify it. If nothing else, this should be enough to get you started in the right direction so that if you have to ask questions of your local gun store, you'll be more informed.