The estimate for the lowest possible orbit has a period of 88 minutes or 6553,6 km semimajor axis radius, or 175 km altitude - if its circular.
In reality though, thanks to physics, you can dip much lower as long as you have eccentricity.
I don't think it would be a smart idea to try this at the Kármán line, other than just out of curiosity / for a short-term very low Earth orbit experiment.
There are actually two interesting questions involved, related to a ride with New Shepard and people talking about "astronauts" in this context: where does space begin, and what or who is an astronaut?
Near space might "begin" at the Kármán line. But you don't operate space stations or space shuttles at that altitude

You are still very close to the mesopause. Sure, the sky is pitch-black and the Earth looks quite curved at that altitude. But it does look so within the upper stratosphere already. Of course conventional airplanes won't work at that altitude, both in terms of lift and propulsion. But atmospheric influence is still significant. Orbital space flight therefore does not take place that low.
As for the term "astronaut", it is a person that sails to or between the stars. Of course nobody does do so yet. But in that sense it actually means a person that is equipped with a vehicle/technology that allows him to travel through and explore space, either in an orbit around Earth or around the Sun or flying on other trajectories to get to another celestial body. New Shepard does not provide any of that. Actually it goes nowhere but just straight up to the mesopause and returns within only a few minutes. They don't even have to deal much with frictional heating during return.
And some people think being weightless does qualify to become an astronaut or has something to do with space. But freefall is a location-independent effect one can create everywhere, even at home on a trampoline. It's not characteristic for being in space, it is just characteristic for being in a freefall condition, which of course happens as soon as you are not on a powered flight anymore in Earth orbit or on your way into deep space. But you can have the same effect on Earth, momentarily (like New Shepard, for about 3 minutes).
Don't get me wrong. It's not that I understate the efforts that were put into New Shepard. It's just always interesting how people get the idea to talk about "astronauts" in this context. For sure I would love to fly with New Shepard. But I would never call myself an "astronaut" afterwards. Because that would mean I am equivalent to Ulrich Walther and every other person that has traveled to and lived in LEO or beyond. I would not be professionally educated and trained to be qualified as a person that travels through, explores and lives in space, aka an astronaut. I would rather consider myself a "near space tourist" or "near space traveler"

I'm also not an airline pilot only because I know how airplanes work and are operated and because I am a passenger every so often.
On the other hand, Earth actually is a natural, self-sustaining space station, with a natural ECLSS, that travels through our solar system. Earth is not an abstract object in the vicinity of space. Space doesn't actually begin or end somewhere - we are just in the middle of space, we are a product of the universe; we actually are the universe. And we even travel through interstellar space by about 2 million kilometers per hour afaik (speed of the milky way). So, aren't we all astronauts?

Anyway, it just always seems strange to me when people talk about "astronauts" in terms of New Shepard or something similar.