Sorry, but as others have said, the LEDs require 7.2 V not 5 V. There are circuits that could boost the 5 V up to 7 V, but they're much bigger, more expensive and more complicated than just putting in a transistor.
Have you considered using SMT transistors like SOT-23 package to keep things small? There are transistor arrays around like the ULN2802A (DIP package), but I would've thought that the flexibility of placing the SOT23 transistors individually would use less space than being constrained by the ULN2802A's pinout.
Comment in response to OP's comment (I needed more characters than I can put in a "proper" comment):
Remember that BJTs amplify /current/, not /voltage/. So, if your BJT has a gain of 100, then passing a 1mA current from base to emitter will allow 100 mA to pass from collector to emitter. (I'm assuming that you're using NPN here).
They don't need to be Darlington pairs; that's just what happens to be in the package.
Actually, looking at the datasheet, you should be using ULN2803A, not ULN2802A.
The ULN2803A already contains the base current limiting resistors, saving you some components, so you just connect the LEDs (with their current limiting resistors) to the outputs, sinking the current through the transistors, like this:
Make sure you connect up the "COM" pin of the ULN2803A to +12V to prevent any backwards current flow.
However, I'd still suggest that you could try a bit of SMT soldering. Get some SOT-23 transistors and 0805 resistors, and try soldering them to some veroboard. The technique I use is:
1. Put a blob of solder down for one pad.
2. Position the component over one pad. Hold it down with your fingernail (make sure that ONLY your fingernail is touching the package). You'll need long-ish fingernails for this. You could use tweezers or a small screwdriver to hold it down, but I find using my fingernail gives much more control.
3. Use the soldering iron to re-melt the first pad of solder, so that the component sinks in. At this point, you'll find out if any other part of your finger is touching the component :)
4. Once the first pad/pin is soldered, the others are much easier, as the component is already held in place. Just don't hold the iron on to any pin too long, or the heat will conduct through the component and melt the solder on the first pin.