If at high temperatures atoms are more intensely interacting with each other or emitted photons that also could make the core vibrate. Is in these circumstances the radioactive material more likely to fission faster? Can this be used to get rid of radioactive garbage?
In the years following the discovery of radioactivity, physicists and chemists (recall that Rutherford was given the Nobel prize for Chemistry!) investigated the effect of heating radioactive substances. They could detect no effect on the activity, and therefore none on the half life. This was interpreted (as soon the atom had been established as a nucleus surrounded by electrons) as evidence that the radiation came from the nucleus.
The argument was – and still is – that even at furnace temperatures (say up to 3000 K) there will be disturbance to the electron configurations but it will be rare for atoms to be totally stripped of electrons, and violent internuclear collisions will be very rare. Only such collisions would be likely to influence the emission of a particle from an unstable nucleus.
At much higher temperatures and densities (e.g. in a tokamak or in a star) violent internuclear collisions will be common, and I'd guess that the half lives of unstable nuclei would be reduced, but this is not, as far as I know, detectable at 'ordinary' terrestrial temperatures.
There are already two good and correct answers. Especially considering that the OP mainly asks about fission processes, these answers capture the main physics. I would just like to point out that there exist decay processes in the nucleus that do get affected by temperature, even at room temperature scale.
A prominent example are the famous Mössbauer nuclei, which feature recoilless gamma-decay. Let us look at a typical example isotope and it’s decay chain. 57Co decays radioactively (indeed by electron capture, which was given as another example in another answer) to 57Fe. What’s cool is that it ends up in an excited nuclear state of 57Fe, which subsequently decays by releasing a gamma-photon.
These transitions are used in Mössbauer spectroscopy and have many applications. One is to study phonon spectra and lattice vibrations, which are strongly affected by temperature.
For example the so called Lamb-Mössbauer factor is often directly temperature dependent, and is in turn directly related to the broadening of the natural line width and hence to the half-life/decay time.
Note that this effect does not come from a direct influence on the nucleus, but from an influence on the decay channels and the resulting nuclear recoil. This explains why the energy scales of the temperature variation do not have to be the nuclear ones.
You seem to be confusing two separate concepts. The half-life of a radioactive isotope gives the rate at which individual atoms will spontaneously decay. The likelihood that a fissile material will undergo a chain reaction is quite different from its half-life.
For most modes of radioactive decay the half-life of a radioactive isotope is independent of environmental factors such as temperature, pressure, chemical bonds, electric or magnetic fields. This has been confirmed by very accurate experiments.
The only known exception is that some modes of radioactive decay that involve the electrons in the atom (such as electron capture) are slightly affected by chemical bonds which may change the shape of the electron shells around an atom. For more details see this Wikipedia article.
What is dependent on temperature (and on many other environmental factors) is the neutron cross section of a fissile material - the probability that a neutron emitted in the decay of one nucleus will interact with another nucleus. This in turn determines whether or not a chain reaction will take place.
The other answers have come up with a few exotic cases where external factors such as temperature can affect some aspects of nuclear processes (neutron capture cross-section). However, the overall answer is no, temperature does not affect half-life of an isotope.
To expand on why there is no effect, consider that (as you mention in your question) what we perceive as temperature is actually vibration of atoms. You can calculate the vibrational energy of atoms at various temperature and you will find that for typical temperatures achieved in chemical reactions, the energies are of the order of several electron-volts (eV). Nuclear reactions, on the other hand, occur at energies of a few mega-electron-volts (MeV).
So nuclear reactions are around six orders of magnitude more energetic than chemical reactions.
However, there is a way to accelerate nuclear decay by adding energy. You just have to add energy at the scale of MeVs. You can do this using an intense particle beam. The idea is theoretically sound, but it has not yet been experimentally developed.
There is a relativistic effect.
According to special relativity, a (relatively) moving clock ticks slower. What this means is that at high speeds, a particle will survive a little bit longer on average before decaying.
At a higher temperature, the particles in a gas will be traveling faster, so they will decay a little bit slower. The effect will be really small until their speeds approach an appreciable fraction of the speed of light.
I have only heard of this effect being observed in particle accelerators and cosmic rays. The theory should hold if you could heat up a gas enough that the relativistic effects became observable (which is difficult, to say the least), but at that temperature you're going to have all kinds of other nuclear effects.