Yes it absolutely is!
I'm writing this to make this clear that your question is squarely on-topic.
Here is an example of a (very experienced) amateur, Scott Tilley, both finding and determining the orbit of a satellite, using (very nice) amateur equipment, (very nice) amateur software, and of course a software defined radio or SDR.
You can see from images there that the Doppler shift versus time, determined from SDR data could be matched to an orbit using a software package which I believe is available to the public (let's wait for more answers on this, I'm not an expert).
There are several challenges of course:
A small, low gain antenna plugged in to a USB SDR with no low noise amplifier (LNA) is not likely to get you anything useful for most satellites even in LEO. You'll need some combination of
- a really good amplifier,
- a small dish or Yagi antenna
- some luck and some patience waiting for the right satellite to pass, which is transmitting in the frequency chunk you happen to be receiving at that moment.
It's even possible you can do this with any two of the three.
And once you've got the hang of it in terms of observing times and frequency band to look, all you'll need is the really good LNA for at least some satellites.
As pointed out in comments above, lots of people track satellites, there are web pages about it and software packages out there as well.
Having an extended recording of frequency (Doppler shift of the unknown carrier frequency) versus time, software would start reverse-engineering the orbit that would produce this curve. It won't be unique, you won't nail the exact orbit with one pass, but you can narrow it down a lot, and with that predict possible times for its next pass.
If you don't have a directional antenna to get an approximate fix on some look angles, this is really going to be quite hard, so I think simply detecting the satellite will be a major achievement.
The images below is from some really fancy software that (I believe) includes using all kinds of available data on existing satellites (including the spy varieties) to help identify the spacecraft who's orbit matches. What you are asking about is orbit determination and I'm sure that software is available as well.
below x3: SDR and orbit analysis of the recently amateur-re-acquired IMAGE satellite. From NASA’s Long Dead ‘IMAGE’ Satellite is Alive!