Two or more different regulators connected to one battery
Yes, two or more linear regulators can be powered from a single voltage source. However, the arrangement shown in the question will not work: A linear regulator requires some voltage headroom over the nominal output voltage, to work.

For the LM7809 / 7812, for instance, the minimum dropout voltage is specified as 2 Volts. Thus, for the 7809, at least 11 Volts input is required. For 7812, at least 14 volts. Instead, a "low drop-out" (LDO) linear regulator such as the LM2940 is preferred, as it would require merely 0.5 Volts additional headroom for operation.  

This adds another issue: The greater the difference between input and output, the greater the heat dissipation by the linear regulator, for any given load current. Thus, with a 14 Volt input on the 7805, with 100 mA of current to load, the regulator will need to waste (14 - 5) * 0.1 = 0.9 watts as heat. This is not a huge figure, no doubt, but will need a suitable heat sink. 

A solution is to use the output of the 7809 / 7812 as input for the 7805. This reduces the dropped voltage at the 7805, although at the cost of increasing the heat dissipation at the 7809/7812.

For dual rail output requirements such as this, a good solution is a low drop-out linear regulator such as the OnSemi CS8161, which has 12 Volt and 5 Volt outputs, and a 0.6 Volt maximum required drop-out voltage - so it can be operated from a 12.6 Volt or higher supply.