do a google for no dig fence panels/posts

Electric fence.

Or the traditional way to make a light duty fence, driven metal T-posts.

Drive them into the ground with this. It's much easier than using a sledge.

Normally you use T-posts with welded wire fencing or chicken wire. This is the sort of fence you'd use to (try to) keep rabbits out of your garden.

I don't see how you can put up a more attractive vinyl or other fence without digging post holes. My first thought was that maybe you could drive in the metal posts, and then put hollow vinyl posts over them. But I can't think of an effective and reversible way to keep the big hollow posts from rattling around on the much smaller metal posts. Maybe somebody else can.
(Links are for illustration. I don't intend any endorsement.)

Any gate that could span 4 feet would probably need a very secure post to attach to (ie. requires digging).

I have your solve for the "no dig". However, my solution is a slight dig that I think you can deal with. Where you plan to have each fence post, you get one of those disposable plastic flower pots. You dig down just deep enough to submerse the empty pot. Maybe 10 inches. Not big deal. Then, you stand your posts up in those pots and support them so they are plumb all the way around. Even better is if you have the fence sections already attached to the posts and you can lift it up and stand the whole thing in the position that you want with the posts sitting in the empty flower pots. Then, support the fence so it will not move out of place and pour cement into those flower pots. This is not like a typical footing but this absolutely will be sturdy enough to hold that low fence you are talking about while keeping the kid from knocking it over.

Flannel Guy DIY - Youtube.

Thanks guys! Especially DavidR and FlannelGuy

I'm not opposed to digging holes for the fence posts, I just have a small yard and was hoping not to disturb the grass too much. But I can just get a post hole shovel if need be. I just saw pretty good reviews on the zippity stuff, and the wam bam has steel poles that are sledge-hammered 3-feet into the ground, which I thought would be plenty sturdy (of course, I take this with a grain of salt, but their site says its as sturdy as regular fence posts.)

Some spots are tight. For example, I build a french drain and sump pit and noticed that there is a narrow space between the gas line and my front porch steps that would probably fit a steel pole, but maybe not a 4x4 and cement.

Jon, like most others on there, I'm not familiar with the Wam Bam no-dig fencing system other than what I see online. It looks interesting, but I personally would only use it for decorative fencing. I wouldn't trust it for a moving gate, or a fence needing to contain a kid or a pet. It MIGHT work, but I wouldn't put my money on it. The post is pounded into the ground 3ft, but if your soil is loose, so will your fence posts and gates.

A traiditonal poured concrete footing will always be the better option for strength and longevity if you can make it work.

Thanks Steve,

Our soil is clay. It's pretty heavy, but, as you said, it's a toss up as to whether it would work. I was interested in anyone had tried with, or without, success.