Do Christians believe the gospels were written by some of 12 apostles?

Conservative views have evidence for why Mark was written by John Mark, while liberal views offer evidence of their own as to why this gospel wasn't written by John Mark. "[T]he so-called two-source theory, [is] accepted today by almost everyone." An Introduction to the New Testament by the late Raymond Brown is worth looking into.

http://www.vatican.va/roman_curia/congregations/cfaith/pcb_documents/rc_con_cfaith_doc_20030510_ratzinger-comm-bible_en.html

The Muratorian Fragment was likely written about 170 CE. It begins after presumably cutting off Matthew and Mark, though it could be reversed with the idea that one was written by an eyewitness, like Gospel of John. Here is the pertinent part:

... at which nevertheless he was present, and so he placed [them in his narrative]. [1] (2) The third book of the Gospel is that according to Luke. (3) Luke, the well-known physician, after the ascension of Christ, (4-5) when Paul had taken with him as one zealous for the law, [2] (6) composed it in his own name, according to [the general] belief. [3] Yet he himself had not (7) seen the Lord in the flesh; and therefore, as he was able to ascertain events, (8) so indeed he begins to tell the story from the birth of John. (9) The fourth of the Gospels is that of John, [one] of the disciples. (10) To his fellow disciples and bishops, who had been urging him [to write], (11) he said, 'Fast with me from today to three days, and what (12) will be revealed to each one (13) let us tell it to one another.' In the same night it was revealed (14) to Andrew, [one] of the apostles, (15-16) that John should write down all things in his own name while all of them should review it. And so, though various (17) elements [3a] may be taught in the individual books of the Gospels, (18) nevertheless this makes no difference to the faith (19) of believers, since by the one sovereign [3b] Spirit all things (20) have been declared in all [the Gospels]: concerning the (21) nativity, concerning the passion, concerning the resurrection, (22) concerning life with his disciples, (23) and concerning his twofold coming; (24) the first in lowliness when he was despised, which has taken place, (25) the second glorious in royal power, (26) which is still in the future. What (27) marvel is it then, if John so consistently (28) mentions these particular points also in his Epistles, (29) saying about himself, 'What we have seen with our eyes (30) and heard with our ears and our hands (31) have handled, these things we have written to you? [4] (32) For in this way he professes [himself] to be not only an eye-witness and hearer, (33) but also a writer of all the marvelous deeds of the Lord, in their order. http://www.bible-researcher.com/muratorian.html; https://en.wikipedia.org/wiki/Muratorian_fragment

In addition Irenaeus wrote this in Against Heresies also about 170 CE in Book III Chapter 1.

Matthew also issued a written Gospel among the Hebrews³³¹⁰ in their own dialect, while Peter and Paul were preaching at Rome, and laying the foundations of the Church. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia. http://www.ccel.org/ccel/schaff/anf01.ix.iv.ii.html

So, yes Christians believe eyewitnesses or their "scribes" wrote the four gospels.

Eusebius of Caesarea was, as far as I know, the first Church historian to comment on the history of the New Testament books we call the "Gospels". Eusebius wrote in the late 3rd and early 4th centuries.

According to Eusebius, the contents of the Gospel of Mark are attributed to the Apostle Peter, of whom Mark (or "John Mark") was a disciple. The 2nd century Church Father, Justin Martyr, refers to the Memoirs (ἀπομνημονεύματα) of Peter in his Dialog with Trypho the Jew. Irenaeus, Origen, and Tertullian confirm the tradition that Mark's Gospel ultimately came from Peter.

Matthew and John were Apostles, so they, of course, were eyewitnesses. Luke was not one of the twelve apostles, but he was believed to have been one of the Seventy Apostles that he himself described in his Gospel. I do not believe that the authorship of these three Gospels was held by the early Church to be from anyone other than the Apostles whose name they bore.

Not all Christians hold to the above early Church explanation. A summary of alternate opinions can be found in a footnote to Eusebius' Church History in Schaff's Nicene and Post-Nicene Fathers (Series 2, Volume 1):

The question as to the real authorship of our second Gospel [Mark's], or rather as to its composition and its relation to Matthew and Luke, is a very difficult one. The relationship of the three synoptical Gospels was first discussed by Augustine (De Consensu Evangelistarum), who defended the traditional order, but made Mark dependent upon Matthew. This view prevailed until the beginning of the present century, when the problem was attacked anew, and since then it has been the crux of the literary criticism of the Bible. The three have been held to be dependent upon each other, and every possible order has found its advocates; a common source has been assumed for the three: the Hebrew Matthew, the Gospel according to the Hebrews (see Bk. III. chap. 25, note 24), our canonical Gospel of Mark, or an original Mark, resembling the present one; a number of fragmentary documents have been assumed; while others, finally, have admitted only oral tradition as the basis. According to Baur’s tendency theory, Matthew (polemically Jewish-Christian) came first, followed by an original Luke (polemically Pauline-Christian), then by our Mark, which was based upon both and written in the interest of neutrality, and lastly by our present Luke, designed as a final irenicum. This view now finds few advocates. The whole matter is still unsettled, but criticism seems to be gradually converging toward a common ground type (or rather two independent types) for all three while at the same time maintaining the relative independence of the three, one toward the other. What these ground types were, is a matter of still sharper dispute, although criticism is gradually drawing their larger features with more and more certainty and clearness. (The latest discussion upon the subject by Handmann, das Hebräer-Evangelium, makes the two types the “Ur-Marcus” and the Gospel of the Hebrews.) That in the last analysis, however, some space must still be left for floating tradition, or for documents irreducible to the one or two types, seems absolutely certain. For further information as to the state of discussion upon this intricate problem, see among recent works, especially Weiss, Einleitung, p. 473 sqq., Holtzmann, Einleitung, p. 328 sqq., and Schaff, Ch. Hist. I. 575 sqq., where the literature down to 1882 is given with great fullness. Conservative opinion puts the composition of all the synoptic Gospels before the destruction of Jerusalem (for the date of Luke, see III. 4, note 12); but the critical school, while throwing the original type back of that date, considers the composition of our present Gospels to have been the gradual work of years, assuming that they were not finally crystallized into the form in which we have them before the second century.

Some interesting points in this are that (1) Mark's Gospel began to be divorced from Peter and associated with Matthew around a century after Eusebius' history by Augustine, who (in Carthage) was somewhat removed from the theological center of Christianity at the time (Constantinople); and (2) the more exotic theories about the origin of Mark's Gospel have arisen only very recently (in Church time, anyway).