I was looking at some latches but I can't seem to have many that just store the value as 8 bit output.
Your problem is that you don't want to 'just store the value', you want it to stay 'on' even after the input value (button) goes back to 'off'.
S/R latches are perfect for this job. All you have to do is connect each switch to the SET input of its latch, and the reset switch to the CLEAR input of all latches.
Logic ICs are produced to meet popular demand. Few circuits that work with 8 bit data need the functionality you are asking for, which is why you are having trouble finding an 8 bit IC to do it. However ICs that do fewer bits are available, such as the 74LS279 which has 4 SR latches. If that is a real problem for you then there are single chip solutions such as programmable logic (eg. ATF16V8) or a small MCU, but these are a lot more complicated to work with. So just accept that you will need two or more chips to create this circuit with TTL logic.
Your circuit also has a number of other problems:-
5 V TTL logic can only withstand 5 V on inputs, not 12 V.
Normal LSTTL outputs can only source ~0.4 mA, but they can sink ~8 mA. Therefore it is better to arrange the LEDs to light up on logic 0 output, with a resistor in series to limit current to less than 8 mA.
You need 'pull down' or 'pull up' resistors on the switches to define the input voltage when they are off.
LSTTL inputs draw about 0.2 mA when pulled low and the input voltage needs to be 0.8 V or less, so the maximum acceptable 'pull down' resistance is ~4 kΩ. However they only draw ~40 μA when pulled up, so if you arrange the buttons to switch to Ground then a much higher value 'pull up' resistance can be used. Switching to Ground will also make the signals compatible with active low inputs.