What argument and/or evidence is there that Jesus sent the disciples out armed?

The directly applicable scriptural evidence for this is Luke 22:35-38, Luke 22:49-50,and John 18:10-11. Let's start with Luke 22:35-38.

"Then Jesus asked them, “When I sent you out without purse or bag or sandals, did you lack anything?” “Nothing,” they answered. “Now, however,” He told them, “the one with a purse should take it, and likewise a bag; and the one without a sword should sell his cloak and buy one. 37 For I tell you that this Scripture must be fulfilled in Me: ‘And He was numbered with the transgressors.’ For what is written about Me is reaching its fulfillment.” 38 So they said, “Look, Lord, here are two swords.” “That is enough,” He answered."

Although carrying swords is part of Jesus conveying the changing landscape to his disciples and warning them more broadly, it is clear from the transaction that He literally wants them to be carrying swords with "That is enough." He is not just speaking figuratively here.

This leads to Luke 22:49-50.

"Those around Jesus saw what was about to happen and said, “Lord, should we strike with our swords?” 50 And one of them struck the servant of the high priest, cutting off his right ear. 51 But Jesus answered, “No more of this!” And He touched the man’s ear and healed him."

So here, after instructing his followers to carry swords, He tells one of them to cease using the sword and instead heals the man hurt by one.

So in Luke, the takeaway message can be read either way. Jesus could be telling people to be prepared to defend themselves (including carrying swords), or He could instead merely be setting up a situation to illustrate that they ought not to hit people with swords. Or perhaps both - they ought to carry swords to defend themselves, but they don't need to defend Jesus at that moment.

Luke 22:52-53 gives us more context re Jesus' response 'No more of this!'.

"Then Jesus said to the chief priests, temple officers, and elders who had come for Him, “Have you come out with swords and clubs as you would against an outlaw? 53 Every day I was with you in the temple courts, and you did not lay a hand on Me. But this hour belongs to you and to the power of darkness.”"

A similar course of events is described at John 18:10-11.

"Then Simon Peter drew his sword and struck the servant of the high priest, cutting off his right ear. The servant’s name was Malchus. 11 “Put your sword back in its sheath!” Jesus said to Peter. “Shall I not drink the cup the Father has given Me?”"

Clearly, St. Peter was carrying a sword, and clearly, Jesus tells him to cease, but also says it's because of something specific to the situation ('the cup the Father has given me').

Again, this can be interpreted either way. Since Luke 22:35-38 seems to be issuing a general imperative that goes beyond the immediate future ("When I sent you out [...] Now, however" seems to be referring more generally to evangelizing work in the future), you can argue He issues a general imperative first, and then overrides it for that specific instance, where the general imperative continues on after He has been arrested. Yet the precedent of interrupting the violence with self-sacrificing love is difficult to ignore, and obviously would have made a mark on the disciples.

It is not clear to me how to harmonize these threads of the narrative.

Looking more broadly, Jesus used a weapon in at least one situation. He uses a whip to drive money changers from the temple.

"When the Jewish Passover was near, Jesus went up to Jerusalem. 14 In the temple courts He found men selling cattle, sheep, and doves, and money changers seated at their tables. 15 So He made a whip out of cords and drove all from the temple courts, both sheep and cattle. He poured out the coins of the money changers and overturned their tables. 16 To those selling doves He said, “Get these out of here! How dare you turn My Father’s house into a marketplace!”" (John 2:13-16, cf. Matthew 21:12, Mark 11:15, Luke 19:45)

This is not the lovey-dovey Jesus that some would like to imagine. He is overturning tables and at least threatening people with a whip.

He speaks using militaristic language at various points, perhaps most obviously at Matthew 10:34.

"Do not assume that I have come to bring peace to the earth; I have not come to bring peace, but a sword."

This is metaphorical language, and is typically understood in a spiritual sense by Christians - but it is also tough, militaristic language, and along with episodes such as the scourging at the temple, can be translated into physical warfare based on spiritual principles.

Of course, these verses have to be balanced by the entirety of the Gospel, as DJClayworth has mentioned in his answer ("love your neighbour," "do unto others as you would have them do unto you," an overturning of the personal principle of "an eye for an eye, a tooth for a tooth," and so on).

Addendum:

Some people argue against the idea that the disciples went out with swords by referring to Matthew 10:16-20.

"Behold, I am sending you out like sheep among wolves; therefore be as shrewd as snakes and as innocent as doves. 17 But beware of men; for they will hand you over to their councils and flog you in their synagogues. 18 On My account, you will be brought before governors and kings as witnesses to them and to the Gentiles. 19 But when they hand you over, do not worry about how to respond or what to say. In that hour you will be given what to say. 20 For it will not be you speaking, but the Spirit of your Father speaking through you."

Given the disciples are described as 'sheep among wolves', and are told that 'when they hand you over, do not worry about how to respond', this suggests they weren't going to be armed or worried about defending themselves.

A major problem with this argument is that Matthew 10 happens chronologically before Luke 22. Matthew 10:9-10 says

"Do not carry any gold or silver or copper in your belts. 10 Take no bag for the road, or second tunic, or sandals, or staff [...]"

Yet this is explicitly what is being modified with Luke 22:35-38.

"“When I sent you out without purse or bag or sandal [i.e., what Matthew 10 is about], did you lack anything?” [...] “Now, however [i.e., unlike what I told you at Matthew 10],” He told them, “the one with a purse should take it, and likewise a bag; and the one without a sword should sell his cloak and buy one."

The purse and bag (as well as the mention of sandals) have clear correlates in Matthew 10. So it's clear that Jesus' advice at Matthew 10 is being modified with Luke 22. So it is not clear whether an argument about how the disciples ought to respond based on Matthew 10 is still applicable by Luke 22.