Probably not. As one moves into the Earth the temperature increases. At 100 km depth, it is 1500 Centigrade; at the bottom of the mantle, at 3000 km depth, it is 5000 Centigrade! Those are too hot temperatures for almost anything!

The answer to your question greatly depends on what kinds of materials scientists are capable of producing in the future. In order to be able to go down to the mantle, we would need materials that are resistant to high temperatures and pressures, which are the conditions one would experience down there.

It is very unlikely, because there isn't a direct connection between the vents of the volcano and the mantle. Theoretically, if you could dig wherever you want you could go into the mantle from anywhere. The mantle is solid, not molten (although it would be molten if it were on the surface; it's only solid because of the pressure it's under).

Interesting question, although I don't know if I can give you a definitive response. So, I'll say maybe (and personally, I think yes!) Here are some of the difficulties you need to overcome if you want to get to the mantle through a volcano:

1) Temperature: Volcanoes are obviously very hot. The lava/magma is usually at least 1000°C (~1800°F, about 4-5 times hotter than your oven when you bake cookies). Just to give some comparison, various engines in airplanes and jet fighters routinely hit 1300°C for prolonged periods of time. However, the problem with our hypothetical volcano scuba machine is that the extreme temperature of the lava would eventually cause our machine and everything inside of it, including any people, to reach the same temperature of 1000°C unless we have some way to cool our machine. So the extreme temperature of the lava requires that our machine:

a) Be made of some material which is able to withstand at least 1000°C. This is not so much a problem.

b) Be able to dissipate the heat that is being drawn from the lava. (a little more tricky).

2) Corrosive Environment: Imagine we are trying to make some lemonade: we add water, lemon, and sugar (which is a solid). However, we mix everything together and realize that there is no solid sugar anymore! The sugar has dissolved in the water. When we try to put our scuba machine into a volcano, the lava will actually try to do the same thing and dissolve our machine. In fact, lava, which is primarily made of silica, is extremely corrosive, meaning that it tends to dissolve most everything readily. In jet engines, we fight this same problem all of the time: sand in the air is inhaled by the engine and melts when it gets to the hottest part of the engine. The molten sand attacks and eats away at everything inside the engine. Without going into detail, there are ways that we prevent this molten sand from destroying our engine. So the extremely corrosive environment requires that our machine:

c) Be able to stop the corrosive attack of the lava. (probably could be done)

3) Pressure: The mantle starts somewhere around 10 km below the surface of the earth. The Mariana's Trench, the deepest part of the ocean, is around this same depth as well and has a pressure of 16,000 psi, over 1000 times the pressure you or I feel at the surface of the earth. It was only very recently in 2012 that the first human was able to be present onboard a vessel which reached the bottom of the Mariana's trench. Now keep in mind that lava is much denser than water (~3 times as dense) and thus you would expect a much higher pressure than that at the bottom of the ocean, somewhere in the neighborhood of 50,000 psi or 3000 times the pressure you or I feel at the surface of the earth. These are extreme pressures. So the extreme pressures require that our machine:

d) Be able to withstand 3000 times the earth's pressure at the surface. (these are some high pressures, I don't know what could withstand this!)

Generally speaking, each of the four requirements is not particularly difficult to overcome; rather, it's the combination of all of them that make the problem so difficult. Nevertheless, technology is a wonderful thing that progresses slowly but surely and I do believe that we will be able to do this in the future!