Wow! Your lighting is way insufficient. If you review some of the threads here you will see that most seedlings are shorter than these. The reason your plants are reaching for the sky is because they are trying to get closer to the light. Plants are a lot like animals, they do whatever necessary to survive and thrive. Your plants are desperately trying to get more light.
As for transplanting, you can do this. Do follow the advice above regarding burying the long stem, and be sure to get more light on the plant, and a circulating fan. The gentle breeze of a fan will strengthen the stem. Nothing like a little exercise to harden the muscles ;>)
In the future, you might want to plant seedlings in colored (red or blue) party cups. When you are ready for transplanting, it is nice to be able to squeeze the sides of the cup to loosen the soil so that the soil and plant will gently fall into your hand so you can move it to larger pot. You might have difficulty getting that fragile (even more fragile than usual because of the lankiness) plant out of the clay pot.