Does a bail-jumper have any recourse from being apprehended with the
help of illegally obtained information?
No. The bail-jumper has no recourse from being apprehended with the help of illegally obtained information for reasons discussed at greater length below in response to another question posed which is somewhat broader.
Say that a person is a bail-jumper, that is, someone who fails to
appear in court after being let out on bail. If they are apprehended
by a bail bondsman with the help of illegally obtained information, do
they have any sort of recourse?
The person apprehended has very little recourse. The relevant case law has held that an apprehension of a bail-jumper by a private sector bail bondsman or a private sector bounty hunter hired by a private sector bail bondsman, is not "state action" and hence not subject to the constitutional protections that apply to illegal conduct by law enforcement officers acting under color of state law. This is true even though bail bondsman attempting the apprehend a bail-jumper is given profoundly broad statutory authority to take actions that would otherwise be illegal for a private person to engage in when doing so, and even though the bail bondsman is, in substance, enforcing a direction of a court which is a governmental agency.
Certainly, nothing equivalent to the exclusionary rule or Miranda or a lawsuit against the bail bondsman under Section 1983 for a violation of the bail-jumpers civil rights would be available. (Also, the exclusionary rule that applies to exclude evidence obtained illegally in violation of the 4th and/or 6th Amendments doesn't operate to prevent a criminal defendant detained by law enforcement from being detained on an outstanding warrant, even if the arrest is based upon illegally obtained information, although if law enforcement did it, the bail-jumper could bring a 1983 actions against the offending law enforcement officers subject to qualified immunity.)
The bail-jumper would probably have a right to sue the company that disclosed the information illegally for breach of a privacy tort if this was done in a timely fashion. But, attorneys' fees can't be recovered in a case like that, the statute of limitations is typically short, and damages that could be awarded would normally not extend to any harm involving the criminal defendant's failure to be successful in bail-jumping. So, ordinarily the damages would be nominal at best.
Likewise, there might be a claim against the bail bondsman for participation in a civil conspiracy with the company that provided the information to commit a privacy tort. But, this has all of the downsides associated with suing the company providing the information, and also, would pose an additional problem: it is quite likely that a suit against the bail bondsman by the bail-jumper for acts occurring while the bail-jumper is jumping bail is either contractually waived by the bail-jumper in a bail bond agreement with the bail bondsman that courts would uphold despite the fact that it arguably involves an intentional tort by the bail bondsman, or would be barred by a bail bondsman's immunity from liability created under an applicable state statute or the common law of that state created by judicial decisions.
Furthermore, in some states, a suit like this by a bail-jumper against the bail bondsman and also against the company providing the information, would be barred under the equitable doctrine of "unclean hands" that bars someone who has engaged in illegal or improper conduct in connection with the claim for which relief is sought from utilizing the courts in connection with that set of facts.
Realistically, probably the best legal strategy for criminal defendants who have obtained private bail bonds would be to bring a class action against the companies that provide the information and the bail bondsmen who have used it, ideally brought on behalf of criminal defendants who are not bail-jumpers as well as those who are bail-jumpers, seeking injunctive relief only to prohibit continuation of this practice prospectively, subject to contempt of court sanctions from the issuing court if the company or bail bondsman defendants did so.