OR gate circuit with feedback to input
There are a number of problems with your circuit.


You have a 1 V power source. This will not turn on an LED, never mind two in series.
Assuming that's a mistake and you have about 5 V supply (based on R5 and R7 being 100 Ω and a current of 10 mA) then when you close SW2 you will put 5 V across D3 and destroy it.
It's not clear what purpose D1 and D2 serve.
You have wired Q1 and Q2 as voltage followers. Logic circuits would normally use some kind of common emitter.








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A NOR and a NOT will do the job.


Q1 and Q2 form a NOR gate. Turning either on will pull the common collector to ground. Note that the transistors will be driven into saturation so they could be within 0.2 V of ground whereas your arrangement could at the very best get to about 0.7 V below supply.
When Q1 is turned in it will turn on Q3. Its collector will be pulled high. This will turn on Q2 and latch.








simulate this circuit

Figure 2. Minimalist version.









simulate this circuit

Figure 3. Analysis of your circuit. Note that there is 0 V to bias Q2. It can't turn on.


  How can I get the base voltage higher than the emitter?


You can't. That's why logic gates aren't designed that way.