Do any denomination teach that Jesus was born in the Autumn?

Bullinger's Companion Bible Apppendix 179: Parallel Datings of the Times of our Lord gives a long and detailed explanation (including calculations of “the course of Abia” (Luke 1:5)) of why Jesus must have been born on the first day of the Feast of Tabernacles, and circumcised on the Last Great Day of that festival (both high sabbaths).

Some denominations accept that Jesus was likely born during the Feast of Tabernacles, and was certainly not born at Christmas, but even if they believe this to be the true date, and understand the Festival as symbolizing God's tabernacling (taking temporary residence) with humanity, they refuse to celebrate it as a birthday.

…
Faced with these historical facts, and God's clear instruction about mixing pagan practices with worship of Him, we can conclude only that Jesus would not keep Christmas today. Christmas is, in fact, an affront to Him.

In the Old Testament, God pronounced a curse on Israel for adopting pagan feast days: "Behold, I will rebuke your seed, and will spread dung upon your faces, even the dung of your feasts; and ye shall be taken away with it" (Malachi 2:3, American Standard Bible). The Scriptures give us no reason to think that God's abhorrence of the celebration of pagan festivals to honor Him is any less today!

— Was Christ Born on Christmas Day? | United Church of God

But they don't regard the date of his birth as doctrinal, and in general reject the idea of celebrating anyone's birthday:

…
Although it is helpful to know the general season of Christ's birth, we do not find early Christians regarding birthdays, Christ's or any other, as significant. Even as late as 245AD, the noted scholar Origen "repudiated as sinful the very idea of keeping the birthday of Christ" (Encyclopaedia Britannica, 11th edition). It is also important to note that Jesus Christ did not mark the anniversary of His birth, or make reference to it in any such fashion, nor did any of the Apostles so much as even mention Christ's birth date or their own.
— Is there good reason to reject December 25 as Jesus' birth date? | Living Church of God

Do any denomination teach that Jesus was born in the Autumn?

It seems that the Restored Church of God does teach this.

Much evidence exists. If God intended for us to celebrate Christ’s birthday, He would have had the scriptures record the time of His birth, as well as the location. The exact date is not known, yet evidence points to an autumn birth.

Luke 2:8 provides the first indication, as flocks were still in the fields at night. The flocks were brought in for the winter season of mid-October to mid-March.

Another indication is the fact that Jesus was born six months after John the Baptist (Luke 1:24-26, 36). The question then becomes, when was John the Baptist conceived? Because Zacharias’ priestly functions were carried out during the course of Abijah, at the time of the angel’s appearance (Luke 1:5-20), John’s conception occurred in mid-June, as verses 23-24 indicate. Thus, he would have been born in about mid-March. Therefore, Christ’s birth had to be in early autumn.

There is one additional clue. At the time of Christ’s birth, the area of Jerusalem was very crowded with visitors, because of the celebration of the Feast of Tabernacles. People took advantage of this journey to Jerusalem to pay their taxes (see Luke 2:1-5), as they were collected at this time of the year. Leviticus 23:24-40 proves that the seventh month was the time in question. When we combine Luke 2:1-5 and John 7:2-10 we see evidence of Jesus’ parents traveling to Jerusalem for the spring and fall Holy Days.

While there is virtually no evidence of Christ being born on December 25, there is abundant evidence of an early autumn birth. The world celebrates Christ’s birth in a winter pagan “season,” when, in fact, our Savior was born during a fall festival season that actually pictures the salvation of mankind.

Is there scriptural evidence that Jesus Christ's birth was in the fall of the year and not on December 25?

Historically speaking the many dates for celebrating Christ’s birth were seen in antiquity.

The first evidence of the feast is from Egypt. About A.D. 200, Clement of Alexandria (Stromata I.21) says that certain Egyptian theologians "over curiously" assign, not the year alone, but the day of Christ's birth, placing it on 25 Pachon (20 May) in the twenty-eighth year of Augustus. [Ideler (Chron., II, 397, n.) thought they did this believing that the ninth month, in which Christ was born, was the ninth of their own calendar.] Others reached the date of 24 or 25 Pharmuthi (19 or 20 April). With Clement's evidence may be mentioned the "De paschæ computus", written in 243 and falsely ascribed to Cyprian (P.L., IV, 963 sqq.), which places Christ's birth on 28 March, because on that day the material sun was created. But Lupi has shown (Zaccaria, Dissertazioni ecc. del p. A.M. Lupi, Faenza, 1785, p. 219) that there is no month in the year to which respectable authorities have not assigned Christ's birth. Clement, however, also tells us that the Basilidians celebrated the Epiphany, and with it, probably, the Nativity, on 15 or 11 Tybi (10 or 6 January). At any rate this double commemoration became popular, partly because the apparition to the shepherds was considered as one manifestation of Christ's glory, and was added to the greater manifestations celebrated on 6 January; partly because at the baptism-manifestation many codices (e.g. Codex Bezæ) wrongly give the Divine words as sou ei ho houios mou ho agapetos, ego semeron gegenneka se (Thou art my beloved Son, this day have I begotten thee) in lieu of en soi eudokesa (in thee I am well pleased), read in Luke 3:22. Abraham Ecchelensis (Labbe, II, 402) quotes the Constitutions of the Alexandrian Church for a dies Nativitatis et Epiphaniæ in Nicæan times; Epiphanius (Hær., li, ed. Dindorf, 1860, II, 483) quotes an extraordinary semi-Gnostic ceremony at Alexandria in which, on the night of 5-6 January, a cross-stamped Korê was carried in procession round a crypt, to the chant, "Today at this hour Korê gave birth to the Eternal"; John Cassian records in his "Collations" (X, 2 in P.L., XLIX, 820), written 418-427, that the Egyptian monasteries still observe the "ancient custom"; but on 29 Choiak (25 December) and 1 January, 433, Paul of Emesa preached before Cyril of Alexandria, and his sermons (see Mansi, IV, 293; appendix to Act. Conc. Eph.) show that the December celebration was then firmly established there, and calendars prove its permanence. The December feast therefore reached Egypt between 427 and 433. - Christmas (Catholic Encyclopaedia)

Literature of Jehovah's Witnesses regularly mentions the Birth of Jesus to have likely taken place in fall.

The Bible historian Luke tells us that John came baptizing in the 15th year of Tiberius Caesar. (Lu 3:1-3) Augustus died on August 17, 14 C.E. On September 15, Tiberius was named emperor by the Roman Senate. The Romans did not use the accession-year system; consequently, the 15th year would run from the latter part of 28 C.E. to the latter part of 29 C.E. John was six months older than Jesus and began his ministry (evidently in the spring of the year) ahead of Jesus as Jesus’ forerunner, preparing the way. (Lu 1:35, 36) Jesus, whom the Bible indicates was born in the fall of the year, was about 30 years old when he came to John to be baptized. (Lu 3:21-23) Therefore he was baptized, most likely, in the fall, about October of 29 C.E. Counting back 30 years would bring us to the fall of 2 B.C.E. as the time of the human birth of the Son of God. [Insight on the scriptures Volume 1 page 1094-1095]

It might look a bit weird that the birthday of jesus is estimated to be in the year 2 B.C.E. More information about that can be found here Watchower 1968 8/15 p. 504 pagragraph 13, 14. Another more recent source article from Jehovahs Witneses on the subject can be found in The watchtower 1999 11/1 page 4-5. In which it is also warned against speculating about dates.

Jesus warned his followers against speculating about dates. He told his disciples: “It does not belong to you to get knowledge of the times or seasons which the Father has placed in his own jurisdiction.” (Acts 1:7) Earlier, Jesus revealed that even he did not then know when God would execute judgment on this wicked system, paving the way for Christ’s Millennial Rule. He said: “Concerning that day and hour nobody knows, neither the angels of the heavens nor the Son, but only the Father.”​—Matthew 24:36.

The birth of Jesus is of course an important historic event. But the fact that it is not crystal clear mentioned in the bible, such as some other days are, could be considered an indication that it is not that essential, and that it would be unreasonable to be dogmatic about it. But of course it is interesting to consider different possible scenarios, perspectives and arguments.

Early Quakers did not observe Christmas nor mark other 'times and seasons'. They believed that no day was more holy than any other, and believed that each day, and all of life, was sacred (Quaker Faith and Practice 27.39 and 27.42), which is not the same as saying they taught Christ was born in Autumn!

However, books have been written by some who believe Jesus was born in September. This was proposed by Ernest L. Martin in his book, The Star that Astonished the World (ASK Publications, 1996). He claims to have worked out the birth of Jesus to be precisely at 7pm on the 11th September 3 B.C.E. Not that I have read his book. Nor do I know if he is speaking for himself or as representative of any Christian denomination.

He is not he the only one either. I've read various papers also arguing for a late September or early October date. Unfortunately, I only kept one, but that was not written by anyone claiming to speak for a Christian denomination. The author was David Hughes, Professor of Astronomy at the University of Sheffield, England. An article appeared in the 1st September 2009 "Weekend" magazine, which was reviewing a BBC2 documentary due to be screened on Christmas Eve that year. Hughes identified the year for the triple conjunction of stars (which seemed to coincide with a little-known census of Quirinius) as several years earlier than 3 B.C.E. The first conjunction was in May, getting the Magi off on their long journey. The second conjunction would fit in with the star leading the Magi to Jerusalem in September, but the third conjunction happened a few weeks later in November - the one that led them to Bethlehem and the house that Joseph, Mary and 'the child' were then living in. This fits in with the view that many Christians have that the Magi never pitched up on the night of Jesus' birth, but some time later, meaning Jesus was born before that November conjunction - September or October being the contending months.

Most Christian denominations are willing to admit that the actual date of Jesus' birth is not stated in the Bible, and they allow leeway for individuals to hold to various opinions on the matter. After all, nobody's salvation depends on stating belief that Jesus was born in December! Just that he was conceived, miraculously, by the virgin, and born to the acclaim of angels, and the wonder of shepherds. It is the acclaim and wonder Christians still hold for that gift of God from heaven that is critically important - irrespective of what month of the year it happened.

One denomination that ties early October in with their condemnation of the date 25 December as the wrong date for his birth is the Jehovah's Witnesses. They have clearly stated their view that it was early October Jesus was born, as these two quotes show:

"The evidence, then is that Jesus' baptism and anointing took place in the fall of the year 29 C.E. Since Jesus was "about thirty years old" at the time of his baptism in 29 C.E. (Lu 3:23), his birth took place 30 years earlier, or about the fall of the year 2 B.C.E." (Insight on the Scriptures Vol. 1, p467)

"Evidently, Jesus was born sometime in October." (Jesus, the Way, the Truth, and the Life p18)

Other answers here show that the Restored Church of God and the United Church of God also have early autumn / the Feast of Tabernacles as the season of Jesus' birth.