According to Wiki there are three species and a genus known as Beach Peas. Discounting the tropical one, that leaves 2 species of Lathyrus and one Genus, Galactia. Which one are you proposing to grow?

Lathyrus japonicus

I don't know this plant personally but pictures show it trailing rather than climbing. But it does have tendrils so I assume it would scramble up a twiggy support if it came across one. It appears to be endangered so you'd need to check the seed source is bona fide and not taken from the wild.

How would I check that?

Contact the seller and ask for the provenance of the seed i.e. is the seed nursery grown or wild collected? They can always lie but if you are in any doubt don't buy.