What is the biblical basis for why the sending of the Holy Spirit was necessary? Why couldn't the disciples begin following the great commission right after Jesus ascended into heaven? Why was the sending of the Holy Spirit necessary?
The disciples could have carried out the Great Commission immediately, but they had already received word from the Lord that they would receive the Holy Spirit prior to this. Jesus had commanded that they wait in Jerusalem until then.


  Act 1:4,6-8 KJV - And, being assembled together with [them], commanded them that they should not depart from Jerusalem, but wait for the promise of the Father,... When they therefore were come together, they asked of
  him, saying, Lord, wilt thou at this time restore again the kingdom to
  Israel? And he said unto them, It is not for you to know the times or
  the seasons, which the Father hath put in his own power. But ye shall
  receive power, after that the Holy Ghost is come upon you: and ye
  shall be witnesses unto me both in Jerusalem, and in all Judaea, and
  in Samaria, and unto the uttermost part of the earth. 


This passage and the Great Commission of Matthew 28 are invariably linked: They both involve going to all the world with an evangelistic mission. While Matthew doesn't record Jesus mentioning the Holy Spirit like this before his ascension, these additional words of Jesus from around the same time in Acts (after appearing to his disciples and before He ascends) makes the connection to the Commission clear. The apostles already had experience going out on their own and bringing the Message with power of the Holy Ghost (Mark 6:12-13), but they  knew to wait, or else Jesus wouldn't have given that timeline: The Holy Ghost comes upon you, You receive power, You will be My witnesses. This is why it was necessary that they wait before following the Great Commission. The second part of your question is why the Holy Spirit is necessary.

The Gospel's effectiveness doesn't come with persuasion, but with power. 


  1Co 1:17-18 KJV - For Christ sent me not to baptize, but to preach the
  gospel: not with wisdom of words, lest the cross of Christ should be
  made of none effect. For the preaching of the cross is to them that
  perish foolishness; but unto us which are saved it is the power of
  God.


and again:


  1Co 2:4-6 KJV - And my speech and my preaching [was] not with enticing
  words of man's wisdom, but in demonstration of the Spirit and of
  power: That your faith should not stand in the wisdom of men, but in
  the power of God.


Without the power of the Holy Spirit blowing, there can be no conversion:


  Jhn 3:6-8 KJV - That which is born of the flesh is flesh; and that
  which is born of the Spirit is spirit. Marvel not that I said unto
  thee, Ye must be born again. The wind bloweth where it listeth, and
  thou hearest the sound thereof, but canst not tell whence it cometh,
  and whither it goeth: so is every one that is born of the Spirit.


I suppose you could keep asking why and way, but at the end of the day, this is just how God works. Were it not for the Holy Spirit, there could be no effective evangelism. There is active agency on His part that is more important than the actual words used.