Can a vigilante perform an arrest?
This depends on whether the vigilante has the power to perform a citizen's arrest. The rules depend on the jurisdiction (and vary from state to state in the US), but generally the power to perform a citizen's arrest is quite limited. It may include the power to 'pre-empt' an offence if the would-be offender has attempted to commit the offence. However, often the power is limited to particularly serious offences (eg. felonies). There may also be a requirement that the crime occurred in the citizen's presence, or that the citizen was unable to contact the police instead.
Is evidence obtained by a vigilante admissible?
In the United States, the constitutional exclusionary rule generally prevents evidence from being admitted if the government obtained it illegally. I answered a general question about the scope of the exclusionary rule here. However, the Fourth Amendment exclusionary rule does not apply to evidence obtained illegally by a private individual: Burdeau v. McDowell, 256 U.S. 465 (1921).
In Burdeau, the defendant's papers had been stolen and turned over to the government, which proposed to present them as evidence to a grand jury. The Supreme Court said:
In the present case, the record clearly shows that no official of the federal government had anything to do with the wrongful seizure of the petitioner's property ... there was no invasion of the security afforded by the Fourth Amendment against unreasonable search and seizure, as whatever wrong was done was the act of individuals in taking the property of another ... We assume that petitioner has an unquestionable right of redress against those who illegally and wrongfully took his private property under the circumstances herein disclosed, but with such remedies we are not now concerned.
However, the exclusionary rule in some states goes beyond the Fourth Amendment. For example, article 38.23 of the Texas Code of Criminal Procedure provides that evidence illegally obtained 'by an officer or other person' is inadmissible. This provision was apparently enacted to deter vigilantes: Bubany and Cockerell, 'Excluding Criminal Evidence Texas-Style: Can Private Searches Poison the Fruit?' 12 Texas Tech Law Review 611 (1981), p 625.
Can such evidence be admitted even if the vigilante is not present?
Assuming that no exclusionary rule applies, evidence obtained by a vigilante can potentially be admitted through the testimony of a police officer or other witness, subject to the rule against hearsay and the question of reliability.
The rule against hearsay means that a police officer cannot give evidence that a vigilante told them that the accused was guilty. Evidence of this kind is not admissible because the accused has no opportunity to challenge the reliability of the source of the information in cross-examination.
The rule against hearsay does not apply when the probative value of the evidence does not depend on the truth of the absent vigilante's assertion. If the evidence provided by the vigilante is really 'damning' then it might fall into this category. For example, a vigilante might provide the police with a weapon that has the accused and victim's DNA on it, or tell the police that incriminating evidence can be found at a particular location.
A police officer can then give evidence that the weapon was tested and found to have matching DNA, or a search warrant was executed and the incriminating evidence was found. There is no admissibility issue here. However, the fact that the police were tipped off by a vigilante who has broken the law and is not present to face court may cause the jury to reject the evidence as unreliable (ie. it could have been planted).