If you collected these seeds from plants growing wild, chances are they would have germinated and grown outdoors if you'd simply sown them straight into a windowbox. Alternatively starting them in seed trays (as per the link below) or flats as you call them there, pricking them out when they had their second leaves into individual small pots, growing them on, hardening off and then planting in your window box was probably easier.
Now the seeds have started producing roots, I'd pot them, carefully, now, into small pots, and grow them on, then hardening off etc ready for planting outside. Handle with care so as not to break the root though, its going to be a fiddly, delicate job...
See comments below - you may need to pot on or up into deep pots more quickly as the plants grow, and you will likely need a deeper pot than a window box for the plants, given they develop a long, thick taproot (assuming it's not a tropical variety, which it probably isn't if you collected the seeds locally).