The doctrine of penal substitutionary atonement does indeed rely heavily on the idea that Christ's death was vicarious, and thus sometimes vicarious atonement is closely associated with penal substitution. However, the adjective vicarious is applied to other theories as well, by both advocates and detractors of penal substitution. Thus, without context, the phrase vicarious atonement is ambiguous because it is frequently used to refer to several distinct theories of atonement.
Vicarious according to proponents of penal substitution
Proponents of penal substitution argue that Christ's death was vicarious, but recognize that they must narrowly define the sense in which the word is used. Charles Hodge first provides a broad definition:
[Vicarious], according to its signification and usage, includes the idea of substitution. Vicarious suffering is suffering endured by one person in the stead of another, i.e., in his place. (ST, v2, 3.6.3)
He then proceeds to argue that Christ's sufferings are vicarious in a particular sense, recognizing that others use the term differently. Louis Berkhof does the same thing:
The Bible certainly teaches that the sufferings and death of Christ were vicarious, and vicarious in the strict sense of the word that He took the place of sinners, and that their guilt was imputed, and their punishment transferred, to Him. (ST, 3.3.4.A; bold added)
Vicarious in other theories of atonement
Berkhof provides two examples of theologians who oppose penal substitution but employ the word "vicarious" in their definitions. He disapprovingly quotes Horace Bushnell, a 19th century liberal theologian, who rejects penal substitution in his book entitled Vicarious Sacrifice:
Here then we have the true law of interpretation, when the vicarious relation of Christ to our sins comes into view. It does not mean that he takes them literally upon him, as some of the old theologians and a very few moderns appear to believe; it does not mean that he took their ill desert upon him by some mysterious act of imputation, or had their punishment transferred. [...] No, but the bearing of our sins does mean, that Christ bore them on his feeling, became inserted into their bad lot by his sympathy as a friend, yielded up himself and his life, even, to an effort of restoring mercy; in a word that he bore our sins in just the same sense that he bore our sicknesses. Understand that love itself is an essentially vicarious principle, and the solution is no longer difficult. [bold added]
The other theory of vicarious atonement that Berkhof mentions is that of "vicarious repentance," of John McLeod Campbell. Berkhof contrasts it with penal substitution:
This theory, while recognizing the retributive justice of God and the demerit of sin, denies the necessity and possibility of penal substitution, and asserts that the work of Christ in behalf of sinners consisted, not in His suffering for them, but in the vicarious confession of their sins.
Finally, the word vicarious is also sometimes applied to substitutionary atonement more generally. Reformed theologian Robert Reymond describes Anselm's view this way:
[Anselm] interpreted Christ's death [...] as a vicarious satisfaction (satis, "enough"; facio, "to do") offered to God the Father as the legal representative of the Trinity for the sins of the world. (ST, Appendix D, emphasis in original)
We have seen that both proponents and opponents of penal substitution apply the word "vicarious" to other theories of atonement than penal substitution:
- Bushnell: Christ suffered sympathetically
- Campbell: Christ vicariously confessed sin
- Anselm: Christ vicariously satisfied God's honor
Thus, it's perhaps best to understand vicarious atonement as a hypernym of penal substitution and these other theories. Vicarious is a useful adjective, and perhaps most commonly associated with penal substitution, but without context it is insufficient to specify a particular atonement theory.