We see in the Bible, especially in the gospels, that Jesus and His disciples healed crowds of people. Certainly God is able to heal anyone of anything – even to the point of raising someone from the dead. The Bible doesn't mention any particular account of an amputee being healed* but it does say that Jesus healed every manner of disease and sickness (Matthew 4:23, et al). It's very likely that amputees would have been included among those people healed by Jesus and His disciples. So we can answer that Bible affirms that God can and most likely did heal amputees.
Note also the use of the word, “won't.” It implies that God has specifically excluded amputees from receiving His healing. Who's to say that God “won't” heal someone when He has already demonstrated His ability and willingness to heal thousands? And again, though we don't know of a particular instance where an amputee was healed, how can anyone say absolutely that no amputees were among the crowds Jesus did heal? To even ask then why God won't heal amputees is a misnomer. It's a false premise that has already been proven wrong. God certainly could, would, and most likely did heal amputees.
Rather than asking why “won't” God heal amputees, perhaps the video meant to ask why “doesn't” God heal amputees. However, that's still not a fair question to ask since we've already seen that He most likely has. The question might now be, “Why doesn't God heal amputees anymore?” To answer that, I would ask a much broader question: does God heal anyone anymore? I might be at odd with many other Christians but I believe the miraculous healings recorded in the gospels were given only as signs to evidence the authority of Jesus, His apostles, and the early church.
I've written about this before (here and here). I don't want to rehash everything I've already said about this point so I suggest you read my previous posts about this. Let me just say that, before we had the canon of Scripture, God would give His prophets the ability to perform miracles. This was proof that the person was speaking with the authority of God. Jesus Himself said this overtly when He healed a man “sick of the palsy” (Mark 2:4-12). To demonstrate that He had the power to forgive sins, Jesus healed the crippled man. Today, we have the full canon of Scripture. God's revelation is complete. There are no more prophets or apostles and the need for miraculous signs has ended. The Bible is our record of the miracles and the Resurrection is the only evidence we need to know that Jesus has the power to keep His promise of eternal life to us.
O.k., o.k.! I know what some people are thinking: “What about all those faith healers we see on TV?” Actually, I think this would be a good question for them. The people I've seen supposedly healed all had “invisible” ailments. How do I know if some guy really had a bad back or weak heart? I would like to see just one person in these crusades regrow a limb. I'll come out and say it – I think the faith healers are fakes!! I also think all of these prosperity preachers are false prophets. They are charlatans. I should really spend some time on this in the future but let me say just a few things about the matter now. God did not promise us a life of health, wealth, and prosperity. In fact, Jesus told us the opposite. He said, for example, “In the world ye shall have tribulation: but be of good cheer; I have overcome the world” (John 16:33).
As I said in a previous post, I certainly believe God is capable of anything. But what He can do is not necessarily what He does do. I'm the last person who would doubt the ability of God to perform miracles. God has a perfect place prepared for us. A place that is free of the Curse. This earth is not our home and God doesn't intend for us to live here forever. While we're here, there will be death, disease, and suffering – and yes, there will be amputees. But it's only while we're here. In a very literal sense, God will heal everyone who believes in Him.
So why won't God heal amputees? He can, He has, and He will!
[*Added in edit: Luke 22:50-51 does specifically mention Jesus healing the ear of Malchus, a servant of the high priest, after Peter had cut it off with a sword. Thus the Bible does attest to a specific incident where Jesus healed an amputee. So there!]