What is the origin of vicars/priests wearing the white dog collar?

First of all, let us deal with the origins of the term "dog Collar".

The term "Dog Collar" is a nickname of Anglican origin for the neckband of a clerical shirt or a Roman Collar (of the Catholic Church) in the UK.

Dog Collar

An Anglican nickname for the collar that accompanies a neckband shirt—it actually does look something like a flea collar, when you think about it!

Wikipedia has this to say about the "dog Collar":

In the United Kingdom (and other British-influenced countries, such as Canada), clerical collars have been informally referred to as "dog collars"since the mid-nineteenth century. The term Roman collar is equivalent to "clerical collar" and does not necessarily mean that the wearer is Roman Catholic.

The clerical collar seems to be of Protestant origin:

Neckband Shirt

Clergy shirts are Protestant in origin. The Rev. Dr. Donald McLeod of the Church of Scotland invented the neck-band style. (The Church of Scotland is Presbyterian.) Protestant clergy had been wearing white preaching bands for quite some time; McLeod combined them with the detachable collar that was in use at the time. The Roman Catholic Church did not adopt them as streetwear for clergy until later. They modified Rev. McLeod’s design into the tab-collar style.

This article is also interesting:

Where did the clerical collar originate from?

The collar has always been and still is the dress code for Protestant preachers and Lawyers in Europe. In days past, these individuals wore black and chose a white sweatband (cravat) to wear around their necks for the purpose of riding on horseback. This became personified in the UK by John Wesley, founder of the Methodists, who rode on horseback the length and breadth of England preaching the Bible. It was also the normal mode of dress for the Protestant Churches of Europe and it was not until the turn of the eighteenth century that the Catholic Church adopted it also. It was never a Roman Catholic style of clergy dress code before that time and not one picture of a Pope or member of the clergy can be seen wearing one. Sadly today most fundamentalist Protestants and particularly those from other nations erroneously think that the clerical collar is a Roman Catholic instituted style of dress code for their clergy and that Protestants should not wear it, because it represents the Roman Catholic Church, "religion" and "tradition". This type of thinking is wrong. The clerical collar is a Protestant clergy dress code.

Are there any scriptural basis for the use of a clerical collar? In this article, The true origins of the Clerical Collar, it has a long section on the scriptural importance of the collar:

Today, when you look at the clerical dress of the majority of religions, you will see that the leadership attire is very similar. The adaptations in headgear may be different but the style of robe and neckband are ostensibly the same. Because the Judeo-Christian faith is born out of middle eastern customs the origin of Christian clerical attire can be narrowed down to a very definite style.

The thirty ninth chapter of the book of Exodus describes in detail how the Lord commanded Moses to make "the garments of ministry". Again in the book of Leviticus in the eight chapter and verse thirteen, tunics were brought for ministry. The most abominable thing to God is for someone to minister to Him in a secular dress code, using the system of the world's style of attire to flaunt in worship before Him. The spiritual significance of the garments, or covering, before God when conducting Worship, is of absolute importance and reverence in His presence and will affect His presence.

For more information read the entire section on this subject: A distinct dress code is Scriptural!

A Catholic seminarian wearing a cassock with a clerical collar

The terms “Roman collar” or “Roman shirt” refer to style, not origin. Clergy shirts are Protestant in origin. The Roman Catholic Church did not adopt them as streetwear for clergy until the 19th century.