They usually cut the leaves near the base. You can cut them without hurting the plant. I would leave the stalks & collect the seeds. Once you have the plant you have seeds, but trying to find fresh seeds may be difficult. You also need to figure out whether more than one plant is needed for viable seeds.
The leaves are not themselves eaten  just as a flavoring, so tough texture does not matter

I beg to differ. I frequently enjoy eating culantro leaves especially in Vietnamese Phó (a soup). My Puertorican wife also uses culantro in various recaos and salsas. The leaves are not tough!

I also disagree regarding eating culantro leaves. It's the leaves that are the part used. In Thailand you buy them in the markets, or harvest them, when the leaves are about 10 inches long. They are grown in filtered light but primarily shade. And don't leave the seeds on! Flowering and seed setting stops leaf production. If you leave the seed on the plant, it really will give you tough, strong flavored, undesirable leaves. Cut the leaves often for best flavor. Just like any herb, the more you harvest it, the better the flavor. The same with Vietnamese cilantro, basil, etc. Harvest often, keep the flowering cut out and you will have excellent flavored herbs.

Wow, does anyone have any culantro seeds to trade? I would love some.

Where on the plant are the seeds located? On the flower itself? I only see the leaves and the flower stalk.

Culantro is a species of Eryngium, aka Sea Holly. The seeds will form after the flowers have opened, been pollinated and have faded. If you fear missing the seeds you can put a paper or fabric bag over the flower head after it starts to fade and the seeds will be caught and retained in the bag.