apostolic succession of continuing Anglican churches

None is needed. Everyone in the ancient world thought that the claim to be able to teach specialized knowledge required some kind of certification from an authorized teacher (a fact which caused a lot of problems for the Apostle Paul, as two of his genuine letters - Galatians and Philippians - make plain, and which the NT as a whole tries to smooth over by asserting that Paul was endorsed by James). This basic reality is not much different today.

 

That's a very salient question, @Rexlion. Scripture records the ordination of the Apostles in multiple places.

Matthew 10:1-6
"And when he had called unto him his twelve disciples, he gave them power against unclean spirits, to cast them out, and to heal all manner of sickness and all manner of disease. Now the names of the twelve apostles are these; The first, Simon, who is called Peter, and Andrew his brother; James the son of Zebedee, and John his brother; Philip, and Bartholomew; Thomas, and Matthew the publican; James the son of Alphaeus, and Lebbaeus, whose surname was Thaddaeus; Simon the Canaanite, and Judas Iscariot, who also betrayed him. These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: But go rather to the lost sheep of the house of Israel."

Mark 3:13-19
"And he goeth up into a mountain, and calleth unto him whom he would: and they came unto him. And he ordained twelve, that they should be with him, and that he might send them forth to preach, And to have power to heal sicknesses, and to cast out devils: And Simon he surnamed Peter; And James the son of Zebedee, and John the brother of James; and he surnamed them Boanerges, which is, The sons of thunder: And Andrew, and Philip, and Bartholomew, and Matthew, and Thomas, and James the son of Alphaeus, and Thaddaeus, and Simon the Canaanite, And Judas Iscariot, which also betrayed him: and they went into an house."

Luke 6:13-16
"And when it was day, he called unto him his disciples: and of them he chose twelve, whom also he named apostles; Simon, (whom he also named Peter,) and Andrew his brother, James and John, Philip and Bartholomew, Matthew and Thomas, James the son of Alphaeus, and Simon called Zelotes, And Judas the brother of James, and Judas Iscariot, which also was the traitor."
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Nowhere do any of these passages reveal that Our Lord laid hands on the men he chose as Apostles in giving them their apostolic charism. But it should also be noted that nowhere does it expressly state that the laying on of hands were not used. The Scriptures are merely silent on that point. Keep in mind absence of evidence is not necessarily evidence of absence.

It is also important to note what the Apostles did to call Matthias to be a new apostle to replace Judas Iscariot among theur ranks.

Acts 1:16-26
"Men and brethren, this scripture must needs have been fulfilled, which the Holy Ghost by the mouth of David spake before concerning Judas, which was guide to them that took Jesus. For he was numbered with us, and had obtained part of this ministry ... For it is written in the book of Psalms, Let his habitation be desolate, and let no man dwell therein: and his bishoprick let another take. Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us, Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection. And they appointed two, Joseph called Barsabas, who was surnamed Justus, and Matthias. And they prayed, and said, Thou, Lord, which knowest the hearts of all men, shew whether of these two thou hast chosen, That he may take part of this ministry and apostleship, from which Judas by transgression fell, that he might go to his own place. And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles."
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Here again, there is no mention of the laying on of hands. But then, there is no mention of anything being done physically, ceremonially, or symbolically to effect his taking on the mantle of apostleship. Of course, that doesn't mean there wasn't any ceremony, just that St. Luke didn't write it down.

But we must also recognize that the Church's definition of made an Apostle was changing during the New Testament Era. St. Paul is clearly acknowledged as an Apostle, though, as far as we know he was not counted among the twelve nor was he a witness to Our Lord's earthly ministry or His resurrection as was the requirement when choosing Matthias.

What is interesting, though, is that St. Paul did receive the laying on of hands. First by Ananias, acting on message from the Lord that Paul was "chosen [past tense]...to proclaim my name to the Gentiles and their kings and to the people of Israel". Keep in mind Ananias of Damascus was neither an Apostle nor a bishop, as far as we know. Yet, when he laid hands on Paul, Paul's sight was restored and he received the Holy Ghost (See Acts 9). Afterward, in Antioch, Paul and Barnabas are set apart and had hands laid upon them prior to being sent out on a mission to Cyprus (See Acts 13:1-3). Again, this was not done by an Apostles but by Antioch's "prophets and teachers". Paul spent time with certain Apostles in Jerusalem early in his ministry and again 14 years later. Neither time was Paul "made an Apostle" by the Twelve, either through the laying of hands or any other method. Instead, Paul is emphatic, "I did not go to anyone for advice, nor did I go to Jerusalem to see those who were apostles before me. Instead, I went at once to Arabia, and then I returned to Damascus. It was three years later that I went to Jerusalem to obtain information from Peter, and I stayed with him for two weeks. I did not see any other apostle except James, the Lord's brother" ( Galatians 1).

Then he says, "fourteen years after I went up again to Jerusalem with Barnabas, and took Titus with me also.
And I went up by revelation, and communicated unto them that gospel which I preach among the Gentiles, but privately to them which were of reputation, lest by any means I should run, or had run, in vain." It was at this time that Paul rebuked Peter and also where he declares himself the Apostle to the Gentiles just as Peter was the Apostle to the Jews (Galatians 2).

Thus, it appears to me that, at least from the biblical record, hands-on ordination from an existing Apostle or bishop was not an absolute requirement to be an Apostle.

 

There is a small crowd who dispute the authenticity of St. Matthias' apostolate. They tend to make two principle arguments: 1. casting lots is not an appropriate way to choose a successor 2. the book of Revelation says that there were only 12 (of course, they always choose St. Paul as the 12th over St. Matthias). So we have a moral quibble (casting lots is gambling in this perspective) and a literalist reading of Revelation. The same methods apply to a number of other viewpoints of ordination.

 

That's interesting that folks would have problems with casting lots. It seems little different from the high priest consulting the Urim and Thummim on behalf of Saul and David.

 

Well, casting lots seems a bit similar to "putting out a fleece," both of which are seeking God's will by looking for a physical sign from Him rather than trusting the Spirit of the Lord to guide the choice. (Some would say, if you keep putting out fleeces, sooner or later you're going to get fleeced!) In the NT era particularly, seeking a physical sign is inferior to seeking God in prayer (and perhaps in fasting, too) for believers in whom the Holy Spirit dwells.