Is it the quantum tunneling that causes the charges to go from emitter to collector rather than to base in BJT?
This is not the case. 

What is going on is the following:

Charge carriers move along the electric field gradient of the collector-base junction. They then have some amount of speed, traveling in that direction. As the carriers have some lifetime (average time until a electron-hole pair recombine), this allows them to drift on. If they go fast enough, they can move into the emitter. Only a small fraction doesn't make it, and this fraction causes the base current to flow. 

A shorter base region will improve how much of the carriers make it (better \$\beta\$). A crystal lattice with less defects will do the same (as it increases the lifetime, and thus the amount of carriers that make it).