How can the theory of emptiness be true and yet the self still transmigrates and takes rebirth? The Theory of "Emptiness" is the concept that all phenomenon are empty of inherent existence. Something has the illusion of existence when the right causes and conditions arise. Example: there is no inherently existing chicken soup. You only get chicken soup when you put together a dead chicken, hot water. Veggies etc. in this theory, you illusory sense of self comes from the right causes and conditions, parents, a body, a brain etc. following this logic, upon death, the self should simply cease to exist because that which made you a self has dissolved and since you do not inherently exist you should just vanish, and yet the Buddha is clear we take rebirth. This is illogical and makes no sense. Perhaps this is why the Buddha never taught a theory of emptiness. This idea is nowhere in the entire Pali Canon.
Not all forms of Buddhism define emptiness in the way you describe; in Theravada Buddhism, for example, emptiness mainly means "empty of self":


  “katamā cāvuso, suññatā cetovimutti”?
  
  “idhāvuso, bhikkhu araññagato vā rukkhamūlagato vā suññāgāragato vā iti paṭisañcikkhati — ‘suññamidaṃ attena vā attaniyena vā’ti. ayaṃ vuccatāvuso, suññatā cetovimutti”.
  
  "And what, friend, is liberation of mind through emptiness?"
  
  "Here, friend, a bhikkhu, gone to the forest or gone to the root of a tree or gone to an empty dwelling reflects thus: 'empty is this of self or what belongs to self.' This one calls, friend, liberation of mind through emptiness."
  
  -- MN 43


Being empty of self is different from not existing; it simply means that, from the point of view of experience, that which comes into existence does so from non-existence, and subsequently returns to non-existence; it has no existence outside of the single moment:


  “aniccā vata saṅkhārā, uppādavayadhammino.
  uppajjitvā nirujjhanti, tesaṃ vūpasamo sukho”ti.
  
  "Impermanent indeed are formations, of a nature to arise and fall.
  Having arisen, they cease, their tranquillizing is happiness."
  
  -- DN 16
  
  “yaṃ kiñci samudayadhammaṃ sabbaṃ taṃ nirodhadhamman”ti.
  
  "Whatever is of a nature to arise, all of that is of a nature to cease."
  
  -- DN 3
  
  yasmā ca kho, ānanda, suññaṃ attena vā attaniyena vā tasmā suñño lokoti vuccati.
  
  Because it is, Ananda, empty by way of self or what belongs to self, therefore 'the world is empty' is said.
  
  --SN 35.68


There is only one example in Theravada Buddhism that I know of that is cited as support for the idea of emptiness being related to non-existence - in the Patisambhidamagga:


  katamaṃ vipariṇāmasuññaṃ? jātaṃ rūpaṃ sabhāvena suññaṃ.
  
  What is emptiness due to change? Born form is empty by way of intrinsic existence.
  
  -- Paṭis. XX.5


The commentary to this passage refutes the idea that this refers to non-existence, however:


  'Void of individual essense': here sabhāva is sayaṃ bhāvo; arising of itself is the meaning. Or sabhāva is sako bhāvo; own arising. Because of existence in dependence on conditions there is in it no essence by itself or essence of its own, thus it is "void of individual essence".
  
  ...
  
  But if someone should say: "Own essence is individual essence; it is void of that individual essence. ... Because of the non-existence of any idea whatever it is the non-existence of materiality that is expressed by the words 'born materiality is void of individual essence.'" It should be argued thus: that being so, the words 'born materiality' would be contradicted; for what is devoid of arising is not called 'born'. ... And the word 'void' for what is non-existent contradicts the Blessed One's use of it above for the 'world' and also the words of the books of logic and linguistics; and it contradicts many logical arguments.
  
  -- Path of Discrimination, p. 362 (Ñāṇamoli, trans)


So, while to say that "the Buddha never taught a theory of emptiness. This idea is nowhere in the entire Pali Canon." is clearly false (and should really be removed from your question), there are ways of understanding emptiness that don't contradict existence.

Further, the Buddha taught non-self; he didn't precisely teach "rebirth" - he never actually used a word, that I know of, that implies such an idea. He taught dependent origination; that mental formations lead to further becoming. It is pretty clear from reading the earlier texts (e.g. those cited above) that experiences arise and cease, being replaced by new experiences. This occurs even at the moment of physical death, hence "rebirth", or more accurately "birth".

In summary, emptiness can be understood to refer to absence of self (i.e. something eternal or immutable) and rebirth isn't really a Buddhist concept. Birth comes about because of mind states in one existence leading to future mind states in another existence - the separation between existences is merely conventional.