I have one of those 4ft buried frost free spigots by my shop.  I usually leave it turned on, with a hose running to a small stock tank for my goats that are in the field next to my shop.  The float in the stock tank regulates the water...turning it on and off as needed.

Lately, I've noticed that the ground all around the spigot is wet a soggy.  I put my ear to the spigot and when it's turned ON (but no water flowing, stock tank is full so float valve is closed)....I can hear water hissing out underground.  When I turn the spigot off, the water hissing sound stops.  I left it off for a few days, and the soggy ground dried up.  Turned it back on, and a few hours later the ground is wet again.

So here is what I think is going on.  When the spigot valve is closed...everything is fine, no leaks.  But when I turn the spigot on....the water is wanting to flow out through the weep hole at the bottom of the spigot. I think the water pressure is bypassing a seal or something and making it's way back to the weep hole.....which is only supposed to be exposed to the water at the valve when you turn the spigot off, so the water can drain out...which is does.

Any ideas on how to fix this?  Does the same rubber plunger that is in the valve assembly at the bottom of the spigot also seal off the water from going out the weep hole when the spigot is turned on?

First thing I would do is figure out who manufactured it.
Woodford for example has a very nice website with troubleshooting info and replacement parts.

To answer your question either you have a hole in it or the plunger is bad. The plunger seals the drain hole when the hydrant is on.

I've got a couple of them at the ranch and always wondered if they could be repaired... your post prompted me to look it up.
Several videos are on Google and in 2 or 3 minutes I learned how to repair them. There is a rubber plunger at the bottom that is replaceable.

That would be nice.  Last time I had to fix one it took 6 hours to dig it up to repair.  Course that was 25 years ago.

Easy fix. Most likely the plunger is worn. You don't have to dig it up to fix it. Order correct plunger from the manufacturer. As the valve is opened the plunger allows water in while sealing weep hole. When closed it stops water and opens weep hole. You will need two pipe wrenches, probably 24s, to hold backup and unscrew the top from the pipe stand. Everything simply pulls out. If you don't have a good backup hold, you could damage the connections underground.

There is also a possibility that the pipe somewhere above the valve is compromised but I would replace the plunger before I started digging.

You can replace the plunger, but if it is an older unit, what the plunger seats against may be worn or coroded, and you may never get a good seal.  Also, buy a 90 degree elbow for your weep hole, this helps keep it clear, and don't forget lots of gravel around there as well.  I tried to repair an hydrant on the farm and just ended up getting a new Woodford with 48" bury depth.

The packing tension is adjustable on the top in the Woodford IOWA spigots.  Start there.

Adjust the rod length with the pair of nuts at the shutoff handle.
If that doesn't do it, pull up the plunger.
Replace the leather seats.

So with the spigot on and no water flowing because the float has it shut off (full tank) could the pipe ruptured due to being frozen and full of non moving water?  Much like leaving a garden hose connected to a house faucet.

The plunger, probably draining out weep hole before it can get up to hose bib which is another sign that the plunger is worn.