Scientists routinely use the onboard transponders of deep space missions to return signals from the Deep Space Network in order to determine their distance from earth. This is briefly described in this Scientific American article.
Since the furthest object ever visited by a deep space mission is Ultima Thule by the New Horizon's spacecraft, it stands to reason that this is the " farthest distance to a solar system object that's been determined by spacecraft transponder". At flyby the distance from the Sun was around 6.6 billion kilometers, according to the above article, or 6.64 billion kilometers from the earth according to this article.
The JPL Horizons web interface can be used to obtain state vectors for Ultima Thule.
The data returned contains the following information in the header:
Revised: May 21, 2019 486958 (2014 MU69) 2486958
This pre-computed trajectory is consistent with the New Horizons spacecraft
Kuiper-Belt extended mission, with the reconstructed 3537.7 km flyby of
2014 MU69 on 2019-Jan-1 @ 05:34:31 TDB (05:33:22 UTC).
Trajectories were provided by the New Horizons mission planning team at SWRI
and are fits to internal flight-project data that has not been made available
outside the flight project.
So it seems that Ultima Thule's orbit has been improved by using reconstructed flyby data from the New Horizons spacecraft, which itself would be constructed using range-rate data from the spacecraft's transponder.
Therefore to the question
Farthest distance to a solar system object that's been determined by spacecraft transponder?
The answer is 44.256 AU from Earth to Ultima Thule (2014 MU69) on
2019-Jan-1 @ 05:34:31 TDB (05:33:22 UTC):
Date__(UT)__HR:MN R.A._(ICRF/J2000.0)_DEC APmag S-brt delta deldot S-O-T /r S-T-O
2019-Jan-01 05:33 19 08 36.78 -20 34 44.1 n.a. n.a. 44.2560922822249 2.9001642 6.1051 /T 0.1431
delta deldot =
Range ("delta") and range-rate ("delta-dot") of target center with respect
to the observer at the instant light seen by the observer at print-time would
have left the target center (print-time minus down-leg light-time); the
distance traveled by a light ray emanating from the center of the target and
recorded by the observer at print-time. "deldot" is a projection of the
velocity vector along this ray, the light-time-corrected line-of-sight from the
coordinate center, and indicates relative motion. A positive "deldot" means the
target center is moving away from the observer (coordinate center). A negative
"deldot" means the target center is moving toward the observer.
Units: AU and KM/S