Your focus seems to be on "morality", but the more interesting question would be that of "agency".
The thing is if you define morality as a code of conduct (descriptive or normative, by society, a group or an individual or whatnot), then the implied assumption is that you're a) dealing with agents that can do conduct and b) that they understand the rules.
So if you're dealing with inanimate objects that can't do anything or with animate beings that don't understand what they do or what you want them to do. Then you can assign labels to them, but these labels would be for you not for them. Like you'd tell yourself that an object or behavior is good or bad, but your assessment would not have any normative consequence for the thing that you describe.
Like you could look at a poisonous substance and say "bad", but the substance has no possibility to act upon your judgement, it lacks agency.
Like say a wooden spoon is misused for a beating. Is it then "good" because it's a useful tool or "bad" because it's a weapon that inflicts harm? Or does it come down to the object itself in the first place or isn't it more about the person wielding it or the person interacting with it.
So in practical terms it's not a description of the thing itself, but rather about the interaction of an agent with the thing. The thing just is. Of course you can argue that it's existence itself is good or bad, but that doesn't change the morality of the thing, it just reflects your preferences towards the thing.
You could do some mental gymnastics and pretend that the thing is an agent and for example argue that a stick is good because it tries to help people to the best of it's abilities (it just doesn't have any abilities). Or that idk a radioactive substance is bad as it does harm to living things in it's environment. But it's purely descriptive, on their own they couldn't change that behavior and act "moral" or "immoral".
Now depending on your definition of morality that could already suffice, like if morality is just acting according to a code of conduct and you happen to do so by accident or happen to fail to be able to do so, equally by accident. Then you'd be moral or immoral. But that kinda feels pointless. It's like looking at a car driving in one direction and arguing "I command you to further drive in that direction". Sure it will probably do so unless it comes towards a junction but it's not really because of your command, it's because it would have done that anyway.
So if the point of morality is just that things act a certain way and they happen to do so and you really don't care for their reasons to do so, then you could assign a moral label to them.
So with respect to:
Can they still be moral or immoral?
Probably yes (if you really bend the definition to make them to).
With respect to the other question:
Or be held accountable for their actions?
It's more difficult. Because as implied in the beginning, in order to meaningfully hold them accountable for their action, they'd need to have some sort of agency. They'd need to know what to do and deliberately do something else. So there needs to be an understanding of the code of conduct, there needs to be an act that goes against that and there needs to be at least one feasible option to adhere to it.
And with children and animals you're already hitting a problem with regards to understanding the code of conduct in the first place. Like they don't innately do that and it kinda comes down to you teaching them and so their failure to adhere to the rules might actually be your failure to teach them correctly and in an understandable manner.
Legally this is often treated as binary as in "they do or they don't understand and are able to adhere to the rules", but practically the agency to do so is difficult to assess and more on a spectrum and so is the ability to meaningfully hold them morally accountable.
You can of course punish them regardless of their agency and ability, but then you're not really "holding them accountable", you're just harming them because you wanted to or even needed to (poisonous substance for example).
So the question is somewhat how you read the "be" in that question. Is morality a descriptive property of an object, like for example how redness describes the color of an apple. Then yes it can be moral and immoral. Or is it a description of the "character" of a thing, like how a person can have courage. In that case a person can be courageous, but for obvious reasons it wouldn't really make much sense to describe the character of a thing without a character to begin with.